  A totally symmetric set is a subset of a group such that every permutation of
the subset can be realized by conjugation in the group. The (non-)existence of
large totally symmetric sets obstruct homomorphisms, so bounds on the sizes of
totally symmetric sets are of particular use. In this paper, we prove that if a
group has a totally symmetric set of size $k$, it must have order at least
$(k+1)!$. We also show that with three exceptions, $\{(1 \; i)\mid i =
2,\ldots,n\} \subset S_n$ is the only totally symmetric set making this bound
sharp; it is thus the largest totally symmetric set relative to the size of the
ambient group.
