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I was reading an article where it is claimed that

If $G$ is a finite $p$-group with $|G|=p^n$ and nilpotency class of $n-2$ where $n\ge 7$ then $p\le|Z(G)|\le p^2$ and $p^{n-3}\le |G'|\le p^{n-2}$.

I am not sure how these follow, I tried but couldn't succeed. I know from this post that nilpotency class of $G'$ is at most $n-3$ but I can not use it to get the required bound. I am really sorry if I am missing something easy.

If anyone can help me I will be really grateful.

Thanks

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