Is there a finite group which is not the symmetry group of any n-space?

Question

This is a natural follow-up to my previous question, here: A finite group that is not a symmetry group. An $n$-space symmetry is an isometry of $\mathbb{R}^n$. Given a subset $S$ of $\mathbb{R}^n$, the symmetry group $Sym(S)$ is the group of all symmetries that map $S$ to $S$. My question is, is there an example of a finite group which is not isomorphic to any symmetry group in any dimension? I would prefer a minimum cardinality counterexample, if one exists.

Answer 1

It is a result of Cayley that every finite group is isomorphic to a finite permutation group. The permutation domain is arbitrary besides its size, let it be the basis vectors of $\mathbb{R}^n$ for sufficiently large $n$. The group is now isomorphic to permutations of this vector space, which are isometries.