  We consider the action of the symmetric group $S_n$ on the permutahedron
$\Pi_n$. We prove that if $\sigma$ is a permutation of $S_n$ which has $m$
cycles of lengths $l_1, \ldots, l_m$, then the subpolytope of $\Pi_n$ fixed by
$\sigma$ has normalized volume $n^{m-2} \gcd(l_1, \ldots, l_m)$.
