  The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the
least nonnegative integer $n$ such that $G$ embeds in the symmetric group
$\Sym(n)$. We prove that if $H$ is a group then $\mu(G)=\mu(G\times H)$ for
some group $G$ then $H$ embeds in $A\times Q^k$ for some abelian group of odd
order, some generalised quaternion $2$-group and some nonnegative integer $k$.
As a consequence, $\mu(G^{n+1})=\mu(G^n)$ for some nonnegative integer $n$ if
and only if $G$ is trivial.
