Special p-Group

from here

A non-abelian $p$-group $N$ is said to be a special $p$-group if $\mathrm{Z}(N)=\Phi (N)=N^{\prime }$ is elementary abelian.

Recall that $N/\Phi (N)$ is also elementary abelian. Let $V$ and $M$ be the corresponding ${\mathbb{F}}_{p}G$-modules of $N/\Phi (N)$ and $\Phi (N)$. By Lemma 2.8(iv), $M$ is isomorphic to the exterior square of $V$ as ${\mathbb{F}}_{p}G$-modules.

Since $N^{\prime }=[N,N]=\mathrm{Z}(N)$, we have that the nilpotency class of $N$ is $2$.