A question about the names of the non-abelian groups of order $p^3$.

Question

It is well known that given an odd prime $p$, there exist two finite non-abelian groups of order $p^3$: A group with exponent $p$ and a group with exponent $p^2$. The group with exponent $p$ is known as Heisenberg group modulo $p$ and is usually denoted by $H_3(\mathbb Z/(p))$. Does the group with exponent $p^2$ has a special name? If not, does it have a special notation? Thanks.

Answer 1

It may be interesting to mention that the other group $G_p$ of order $p^3$ is the unique Sylow $p$-subgroup of the affine group ${\rm Aff}(\Bbb Z/p^2)$, i.e., the kernel of the homomorphism $$ {\rm Aff}(\Bbb Z/p^2) \rightarrow (\Bbb Z/p^2)^{\times} $$ given by $\begin{pmatrix} a & b \cr 0 & 1\end{pmatrix} \mapsto a^p$.

But of course the name "unique $p$-Sylow subgroup of an affine group" is too long.

For a similar discussion concerning the notation here see

Notations for groups of order $p^3$