Frattini quotient ($G/\Phi (G)$)

The Frattini quotient of a finite p-group is an elementary abelian p-group. Its dimension over ${\mathbb{F}}_p$ is the minimum number of generators of $G$; in particular, it has dimension at least two when $G$ is noncyclic.

A subset generates a finite p-group precisely when its images span the Frattini quotient. Indeed $⟨S⟩\Phi (G)=G$ implies $⟨S⟩=G$, since otherwise a maximal subgroup contains both factors. Minimal generating sets correspond to vector-space bases in the quotient.