Burnside basis theorem ($d(G)={\dim }_{{\mathbb{F}}_p}G/\Phi (G)$)

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.