  Given a permutation group $G$, the derangement graph of $G$ is the Cayley
graph with connection set the derangements of $G$. The group $G$ is said to be
innately transitive if $G$ has a transitive minimal normal subgroup. Clearly,
every primitive group is innately transitive. We show that, besides an infinite
family of explicit exceptions, there exists a function $f:\mathbb{N}\to
\mathbb{N}$ such that, if $G$ is innately transitive of degree $n$ and the
derangement graph of $G$ has no clique of size $k$, then $n\le f(k)$.
  Motivation for this work arises from investigations on Erd\H{o}s-Ko-Rado type
theorems for permutation groups.
