Fix a positive integer $n$ and let $S$ be the set of $n$ by $n$ matrices with entries in $\mathbf{Z}_p$ (the $p$-adic integers) whose determinant is $p$. The group $G:=\mathrm{SL}_n(\mathbf{Z}_p)$ acts freely on $S$ via left multiplication.

Is it possible to write down an explicit list of representatives for the orbits of $G$ on $S$? If so, what is this list? If not, is there an algorithm for computing a complete list of orbit representatives?

For example, for $n=2$, I think that this is basically the "Hecke Operator at $p$" computation, and the $p+1$ orbits of $G$ on $S$ have representatives

$$ \left(\begin{matrix} p & b \\ 0 & 1 \end{matrix}\right),\ b=0,1,\ldots, p-1\ \text{and}\ \left(\begin{matrix} 0 & -1 \\ p & 0 \end{matrix}\right) $$

(sorry if these matrices don't appear correctly in mathjax...)

This is probably a question whose answer is well-known to many people, so I apologize in advance for my ignorance!