Golod-Shafarevich inequality between dimensions of first and second cohomology groups of finite p-group

Statement
This result is a corollary of the Golod-Shafarevich theorem, a result about graded algebras. The statement in the group theory context is as follows: suppose $$p$$ is a prime number and $$G$$ is a nontrivial finite p-group. Denote by $$d$$ the dimension as a $$\mathbb{F}_p$$-vector space of $$H^1(G;\mathbb{F}_p)$$ and denote by $$r$$ the dimension as a $$\mathbb{F}_p$$-vector space of $$H^2(G;\mathbb{F}_p)$$. Then, $$r > d^2/4$$.

Note that $$d$$ is equal to the minimum size of generating set for $$G$$ and $$r$$ is the minimum over all presentations of $$G$$ of the number of relations used in that presentation (is it?). Thus, the Golod-Shafarevich inequality can be stated using the language of generating sets and presentations.