No. It's more like a caste system.
A caste system is a social system. It only says this person is social above or below you. It doesn't say how those people make money. Communism is a political/economic ideology. It definitely orders how money is made/distributed. It also has a type of caste system in party members/nonmemebers. But thats a superficial similarity and not the defining characteristic. Communism may have a caste system but a caste system isn't necessarily communism.
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