No, they didn't. They claimed they did, and then they elevated the party elite to a new upper class and brutally murdered anyone who dared continue to criticise it, including their own party members.
That's misleading. The party elite was not living in much greater luxury than the general Soviet population. The party leaders did not indulge in Saudi-Arabia style opulence. They did kill a lot of people, but that's orthogonal to the question of material equality.
The Soviet Union and its satellites were one of the most egalitarian societies ever seen. What this shows is that equality and lack of socio-economic stratification is not necessarily a good thing. Of course already Marx/Engels were criticised along the lines of: any organisation strong enough to enforce equality will need to be so powerful, so brutal that it will not relinquish power after it has achieved economic equality, leading to all manner of negative consequences.
