Heisenberg group (awaiting review)

Given a prime $p$, the Heisenberg group $\He_p$ is the group of $3 \times 3$ unitriangular matrices over the finite field $\F_p$. Explicitly, it is given by $$\He_p := \left\{ \begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{pmatrix} \mid a, b, c \in \F_p \right\}$$ with the usual matrix multiplication as the group operation. It has order $p^3$, and is one of only two non-abelian groups of order $p^3$. In particular, for an odd prime $p$, it is the unique non-abelian group of order $p^3$ and exponent $p$.