$p$-Sylow subgroup of $S_{p^3}$

Question

Let $p$ be a prime. Prove that $(\mathbb{Z}_p\wr\mathbb{Z}_p)\wr \mathbb{Z}_p$ is isomorphic to a $p$-Sylow subgroup of $S_{p^3}$. Here, $\wr$ denotes the wreath product and $\mathbb{Z}_p$, the cyclic group of order $p$. Please explain in detail.

Answer 1

The structure of the Sylow $p$-subgroups of $S_n$ is indeed known: The Sylow $p$-subgroup of the symmetric group on $n=p^k$ letters is
$$ \underbrace{C_p \wr C_p \wr \cdots \wr C_p}_{k}, $$ that is, an $k$-fold wreath product of cyclic groups of order $p$. See Huppert, Endliche Gruppen I, Satz III.15.3 for a proof.