$\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$.
$\textbf{TRY- }$ If $P=Z(P)$ it is true. Now let $n > 1$, then
If I see $P$ as a nilpotent group and construct its upper central series, it will end , so let it be,
$e=Z_0<Z_1<Z_2<......<Z_r=P$
Now as $Z_{i+1}/Z_i=Z(P/Z_i)$, so if if I take some $x\in Z_2$\ $Z_1$ then $N$={$[x,y]|y\in P$} $\leq Z_1(P)$ and $N \triangleleft P $, so $P/N$ is a group with order $\leq p^{n-1}$.
Now if I let $H=P/N$ then obviously |$H/Z(H)$|$\leq p^{n-1}$.
Now $H'\cong P'N/N \cong P'/(P' \cap N)$ so from here I could finally bring $P'$ atleast into the picture, now |$P'$|=$|H'|.|P'\cap N|$ so $|P'|\leq |H'||N|$.
This is where I am $\textbf{STUCK}$
Now , from here how can I calculate or find some power $p$ bounds on $|H'|$ and $|N|$ so i could get my result.