Lower bound on size of second cohomology group for groups of prime power order

Statement
Suppose $$G$$ and $$A$$ are both finite p-groups for some prime number $$p$$, with $$A$$ an abelian p-group. Suppose $$G$$ has minimum size of generating set $$s$$ and $$A$$ has minimum size of generating set $$r$$. Then, we have the following lower bound on the size of the second cohomology group for trivial group action $$\! H^2(G;A)$$. The group $$\! H^2(G;A)$$ is itself a finite p-group and its prime-base logarithm of order is bounded as follows:

$$\! \log_p(|H^2(G;A)|) \ge rs$$