An extraspecial group of odd order has exponent $p$ or $p^2$, and an extraspecial $2$-group has exponent $4$

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

For a finite group $G$, $\exp (G)=\min \{n\in \mathbb{N}:n>0\text{ and }{g}^n=e\text{ for every }g\in G\}$ (The exponent of a finite group).

For odd $p$, $p_{+}^{1+2n}$ is the extraspecial group of that order with exponent $p$ and $p_{-}^{1+2n}$ the one with exponent $p^2$; at $p=2$ the two are named by their number of solutions of $x^2=1$ (Plus and minus type of an extraspecial $p$-group).

$\mathrm{ord}(g)$ is the least $k\ge 1$ with $g^k=e$ (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

For a group $G$ and a prime $p$, $G^p=⟨g^p:g\in G⟩$ (The $p$th-power subgroup $G^p$).

For odd $p$ there are exactly two extraspecial groups of order $p^{1+2n}$, distinguished by their exponent, which is $p$ for one and $p^2$ for the other (For odd $p$ and each $n\ge 1$ there are exactly two extraspecial groups of order $p^{1+2n}$, distinguished by their exponent).

For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$, separated by the number of solutions of $x^2=1$ (For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$).

There are $n\ge 1$ subgroups of $P$, each nonabelian of order $p^3$ with centre $Z(P)$, forming an internal central product of $P$ (Every extraspecial $p$-group is an internal central product of nonabelian subgroups of order $p^3$).

For every prime $p$ there are exactly two nonabelian groups of order $p^3$; at $p=2$ they are $\mathrm{Dih}(C_4)$ and $Q_8$ (For each prime there are exactly two nonabelian groups of order $p^3$ up to isomorphism).

In $\mathrm{Dih}(C_4)$ the rotation has order four, and in $Q_8$ the element $\mathbf{i}$ has order four ($\mathrm{Dih}(C_4)$ and $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively, $Q_8$ is a subgroup of ${\mathbb{H}}^{\times }$ with eight elements, and $-1$ is its only element of order $2$).

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).

For every finite $p$-group $P$, $\mathrm{\Phi }(P)=P^{\mathrm{\prime }}{P}^p$ ($\mathrm{\Phi }(P)=P^{\mathrm{\prime }}{P}^p$ for a finite $p$-group).