  The minimal degree of a permutation group $G$ is defined as the minimal
number of non-fixed points of a non-trivial element of $G$. In this paper we
show that if $G$ is a transitive permutation group of degree $n$ having no
non-trivial normal $2$-subgroups such that the stabiliser of a point is a
$2$-group, then the minimal degree of $G$ is at least $\frac{2}{3}n$. The proof
depends on the classification of finite simple groups.
