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