Word to describe person who is rich in wealth, but is poor in class?

Solution 1:

Is there a specific word for that?

Yes, yes there is.

vulgarian

a vulgar person, especially one whose vulgarity is the more conspicuous because of wealth, prominence, or pretensions to good breeding.

http://www.dictionary.com/browse/vulgarian

Solution 2:

Nouveau riche (this means someone who recently became rich and thus lacking in taste)

a person who has suddenly risen to a higher economic status but has not gained social acceptance of others in that class

[The Free Dictionary]

Solution 3:

From M-W:

Upstart

a person who has recently begun an activity, become successful, etc., and who does not show proper respect for older and more experienced people.

and Parvenu

a person from usually a low social position who has recently or suddenly become wealthy, powerful, or successful but who is not accepted by other wealthy, powerful, and successful people.

Solution 4:

New Money

While someone mentioned "Nouveau riche", I've literally never heard that used in conversation and would have to look it up to understand its meaning. I think the phase you're really looking for that is commonly used is new money*, which sounds like it's talking about the wealth itself but actually refers to the person:

...refers to the man or woman who previously had belonged to a lower social class and economic stratum (rank) within that class; and that the new money—which constitutes his or her wealth—allowed upward social mobility and provided the means for conspicuous consumption

*which shares the same Wikipedia article with the French term