Is $Z_2(G)$ abelian?

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Let $G$ be non-abelian finite p-group and Let $\frac{Z_2(G)}{Z(G)}$ be elementary abelian group
how we can get $Z_2(G)$ is abelian?
If it's needed we also know $|Z(G)|=p$ and $|Z_2(G)|=p^3$