I suggest you give Native Son a read. Focus especially on the final part. It high lites the fact that a marxist(Max) is happy to say that beneath it all is a class struggle. Wright does a wonderful job of pointing out the fact that Bigger(main black character of the story) must overcome the racial oppression put upon him, this oppression is mainly supported by the upper class thus in lies the problem. Even if we got all lower classes to get along there would still be oppression from the top down which is where it is the hardest to deal with.
You must understand that Marx wanted most minorities to put their struggle second to the overall cause of economic equality through socialism. The problem with this is that, it isn't in most mens heart to put aside a battle they've been fighting all their life.