Let $G$ be a group non-abelian group of order $p^3$, where $p$ is a prime number, prove that:
$\fbox{1}$ $|Z(G)|=p$
$\fbox{2}$ $Z(G)=G'$
$\fbox{3}$ $\frac{G}{Z(G)}\cong \frac{\mathbb{Z}}{p\mathbb{Z}}\times \frac{\mathbb{Z}}{p\mathbb{Z}} $
$G'=\langle \,\{ [x,y] \mbox{: }x,y\in G \} \, \rangle$ commutator subgroup of $G$ where $[x,y]=xyx^{-1}y^{-1}$
Any hints would be appreciated.