An extraspecial $p$-group of order $p^{1+2n}$ has generator rank $2n$

Given: An extraspecial $p$-group $P$ of order $p^{1+2n}$.

For a finite $p$-group $P$, the generator rank $d(P)$ is the common size of a basis of $P\mathrm{/}\mathrm{\Phi }(P)$ (The generator rank $d(P)$ of a finite $p$-group).

A subset $S$ of an elementary abelian $p$-group spans when every element is a product ${\prod }_{s\in S}{s}^{a_s}$ with coefficients in ${\mathbb{F}}_p$, and is independent when such a product is the identity only for zero coefficients; a basis is an independent spanning subset (${\mathbb{F}}_p$-spanning sets, independence, and bases in an elementary abelian $p$-group).

For a finite $p$-group $P$ the following are equivalent: $P$ is extraspecial; $P$ is nonabelian, $\mathrm{\mid }Z(P)\mathrm{\mid }=p$ and $P\mathrm{/}Z(P)$ is elementary abelian; $P$ is nonabelian and $Z(P)=P^{\mathrm{\prime }}=\mathrm{\Phi }(P)$ has order $p$ (Three equivalent descriptions of an extraspecial $p$-group).

An extraspecial $p$-group has $\mid P\mid =p^{1+2n}$ with $n\ge 1$ and $\mid P\mathrm{/}Z(P)\mid =p^{2n}$ (An extraspecial $p$-group has order $p^{1+2n}$ for some $n\ge 1$).

Every finite elementary abelian $p$-group has a basis; every independent subset extends to a basis, every spanning subset contains a basis, and all bases have the same finite size (Finite elementary abelian $p$-groups have bases, basis extension, and a well-defined dimension).

A subset $X$ of a finite $p$-group $P$ is a minimal generating set if and only if the quotient map restricts to a bijection from $X$ onto a basis of $P\mathrm{/}\mathrm{\Phi }(P)$ (Burnside Basis Theorem).

The rule $\bar{a}\cdot x=x^a$ gives every elementary abelian $p$-group its canonical ${\mathbb{F}}_p$-vector-space structure (An elementary abelian $p$-group has a canonical ${\mathbb{F}}_p$-vector-space structure).