There are $p$-groups such that they can not be isomorphic to $G/Z(G)$ for any group $G$. Perhaps such thing may not arises if we replace $Z(G)$ by a term of lower central series. Then my question is the following:
Question: Is there an example of a non-cyclic $p$-group $H$ such that $H$ can not be isomorphic to $G/\gamma_k(G)$ for some $k\geq 2$?
I was feeling that answer should be NO, but I don't know how to prove if it is so.