Fascist political systems were not designed to create a classless society, but a new class system, a "new order," with the fascists themselves (of course!) at the top. Fascists claimed that class differences were not important, but that the important struggles and exploitation were relationships among nations. Some nations were "proletarian nations," exploited by other nations, because of historical accidents or of treachery by some part of their own populations. But a superior nation, with a strong, absolute leader and with a great revolutionary act of will, could reverse that, making itself a new national ruling class over its former oppressors and over the whole world. This would be the "new order."
What reason might an "oppressed nation" have to think that it might really be possible to conquer its neighbors and oppress them in turn? In the 1920's and 1930's, the Fascist nations found historical precedents to convince themselves. For Japanese fascism, the precedent was relatively recent. Only a few hundred years before, their neighbors the Manchu people had made themselves the rulers of China -- that is, all of the world that really mattered from the Manchu point of view in 1600. For Italian fascism, there was the Roman Empire. The ancestors of the Italians themselves had, two millennia before, dominated (almost) all of the world they could reach. For the Germans, the case was more difficult. Their ancestors had once overcome half of the Roman Empire (but then divided among themselves and given most of it up to others) and, a few hundred years later, German knights had dominated many of their neighbors. But that is thin stuff, so the German Fascists claimed that the Germans were really Aryans, who had (they supposed) conquered pretty much everything, just a few thousand years ago. This idea that the clock could be turned back to a time when Our Nation Ruled the World was an important part of Fascism.
Fascists were contemptuous of profit motives, and aimed to subordinate all other purposes (including profit) to the national objectives of victory over other nations. Thus, they rejected market systems in principle, and officially established "planned economies." In practice, they fell even further short of "optimal planning" than the Soviets. (Since the Fascist powers were able to survive for only about 20 years before being conquered by the Soviets and the democracies, perhaps they just didn't have time to get it working). Although terroristic, their "economic planning" was, at the same time, disastrously corrupt. Far from subordinating the profit motive to national purposes, the Fascist system seems in practice to have subordinated terroristic national government to the corrupt profit motives of the Fascist party leaders. This so weakened the Fascist nations that they were defeated despite having very good armies in some cases.
It is hard to say this -- given the evil that Fascism has done in this century -- but perhaps we can conceive a Fascist system that would not be avoidably oppressive, terroristic, or warlike. Perhaps we can conceive of an upper class distinguished mainly by its support for traditions that a large majority support. (For myself, I don't believe it is possible, but let us set my doubts, and perhaps yours, aside). It makes sense that a government of this kind would not rely on the untraditional forces of the market to control production. It might adopt a system of central economic planning, with objectives based on its traditionalist views. Like a hypothetical future socialist system, it might even make central planning work, in the service of a class-based and antisocialist ideal. Many things are possible. Time may tell.
The most dynamic Fascist governments were destroyed by their defeat in World War II. After World War II, the colonial governments of many countries of Africa and South Asia were eliminated. These countries, and the poorer Latin and Caribbean countries of the Western Hemisphere, often saw themselves as the victims of exploitation by other countries -- and their colonial history gave them some reason for that view. Many of these "emerging nations" moved toward a considerable degree of government control of the economy and planning. Many called their system "socialist." In practice, their planning tended to be patterned after that of the Soviet Union, but large parts of the economy remained outside both the plan and the market. Also, some of the governments were far more democratic than the Soviet Union, and even those that were not much more democratic were less powerful. On the other hand, they were even less prepared than the Soviet Union had been for the technicalities of optimal planning. More recently, market-oriented systems in less developed countries have been more successful, and, following that example, socialist "emerging nations" have tended to move toward market systems.