A nonabelian group of order $p^3$ is extraspecial

Given: A prime $p$ and a nonabelian group $P$ with $\mid P\mid =p^3$.

$Z(G):=\{z\in G:zg=gz\text{ for every }g\in G\}$ (The center $Z(G)$ of a group).

A finite $p$-group is a finite group whose order has the form $\mid P\mid =p^n$ (A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$).

If $P$ is a nontrivial finite $p$-group then $p$ divides $\mid Z(P)\mid$ (Every nontrivial finite $p$-group has nontrivial center, in fact $p$ divides $\mathrm{\mid }Z(P)\mathrm{\mid }$).

If the quotient group $G\mathrm{/}Z(G)$ is cyclic, then $G$ is abelian (If $G\mathrm{/}Z(G)$ is cyclic, then $G$ is abelian).

For a finite group $G$ and $H\le G$, $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ (Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$).

If $p$ is prime and $G$ is a group of order $p^2$, then $G$ is abelian (Every group of order $p^2$, for prime $p$, is abelian).

An elementary abelian $p$-group is a finite abelian $p$-group in which every nonidentity element has order $p$ (Elementary abelian $p$-groups).

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).

If $P$ is a finite $p$-group and $H\le P$ then $\mid H\mid =p^k$ for some $k$ (Every subgroup of a finite $p$-group has order a power of $p$).