The paper concerns a remarkable polytope, known as the permutohedron, studied already in 1911 by Schoute. In general, the symmetry group of a polytope, which is the group of its automorphisms, does depend on the considered class of isomorphisms. One can distinguish isometries, affine isomorphisms, face-lattice order isomorphisms, and graph isomorphisms of a polytope. We show that, in case of the permutohedron, these four groups of automorphisms are (group) isomorphic; we also provide their group-theoretical description. Specifically, if $N$ is the basic set, $|N|\geq 3$, then (each of) the automorphism group(s) is the direct product of the group of permutations of $N$ (= the symmetric group $S_{N}$) and of a two-element group generated by central symmetry. The result confirms an unproved statement formulated by Crisman in 2011.
polytope, permutohedron, automorphism, symmetry group
20B25, 52B15