Minimal size of a generating set for presentations of finite groups

Are there any results on the minimal number of generators required to give a presentation of a finite group? More specifically, given a group G, what is the minimal number of generators needed for a presentation of it? No bounds on the number of relations are assumed.

• The direct product of $n$ copies of the cyclic group of order $p$ prime, $\mathbb{Z}_p^n$, cannot be generated by fewer than $n$ elements (via a basic vector space argument). – user1729 Oct 24 '18 at 12:27