Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$. The Heisenberg group $U_{n}$ is the set of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. $U_{n}$ is a subgroup of $G$ of order $p^{\frac {n(n-1)} 2}$, in particular $U_{n}$ is a Sylow $p$-subgroup of $G$. It is well known that the Sylow $p$-subgroups of the group $G$ are conjugate, and every $p$-subgroup $H$ of $G$ is contained in some Sylow $p$-subgroup of $G$. Then there exists $g\in G$ such that $H\leq gUg^{-1}$.
Questions:
How many sylow $p$-subgroup does $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$ have?.
what is the number of conjugacy classes of elementary abelian subgroups of order $p^2$ in $U_{n}$.
Any help would be appreciated so much. Thank you all.