  We study properties of the realizations of groups as the combinatorial
automorphism group of a convex polytope. We show that for any non-abelian group
$G$ with a central involution there is a centrally symmetric polytope with $G$
as its combinatorial automorphisms. We show that for each integer $n$, there
are groups that cannot be realized as the combinatorial automorphisms of convex
polytopes of dimension at most $n$. We also give an optimal lower bound for the
dimension of the realization of a group as the group of isometries that
preserves a convex polytope.
