Symmetric-group embedding at one less than the group order

For $n>1$, a cyclic group of prime-power order $n$ cannot embed in $S_{n-1}$, because a permutation of order $p^k$ requires a cycle of length at least $p^k$. If $n$ is not a prime power, every group of order $n$ embeds in $S_{n-1}$ by combining coset actions on subgroups of two distinct prime orders.