  Given a permutation group acting on coordinates of $\mathbb{R}^n$, we
consider lattice-free polytopes that are the convex hull of an orbit of one
integral vector. The vertices of such polytopes are called \emph{core points}
and they play a key role in a recent approach to exploit symmetry in integer
convex optimization problems. Here, naturally the question arises, for which
groups the number of core points is finite up to translations by vectors fixed
by the group. In this paper we consider transitive permutation groups and prove
this type of finiteness for the $2$-homogeneous ones. We provide tools for
practical computations of core points and obtain a complete list of
representatives for all $2$-homogeneous groups up to degree twelve. For
transitive groups that are not $2$-homogeneous we conjecture that there exist
infinitely many core points up to translations by the all-ones-vector. We prove
our conjecture for two large classes of groups: For imprimitive groups and
groups that have an irrational invariant subspace.
