What is the state-of-art of the following question?
Let $p$ be any prime number. For any finite $p$-group $G$, let $r_G$ denote the minimum number of defining relators in all presentations of $G$ over minimal generating sets of $G$.Let $R_G$ denote the dimension of the second cohomology group of $G$ mod $p$.
Question: Is there a finite $p$-group $G$ such that $r_G > R_G$?
Has it been checked for all $2$-groups of order at most $2^{10}$ or other small $p$-groups which we know them by the small group library of GAP?
Is a possible example studied to have certain properties? e.g. must it be of nilpotency class at least $3$?
