In a joint work with Manuel Delgado (FCUP), I have developed a GAP Packages to calculate Core-free subgroups of a given group and to provide faithful transitive permutation representations of groups.
This package also has tools to build permutation representation graphs, which are useful in understanding the action of groups.
You can find the GAP package "CoreFreeSub" available in
https://github.com/CAPiedade/CoreFreeSub