Finite Permutation Groups with Few Orbits Under the Action on the Power Set

We study the orbits under the natural action of a permutation group \(G \subseteq S_n\) on the powerset \(\mathscr{P}(\{1, \dots , n\})\). The permutation groups having exactly \(n+1\) orbits on the powerset can be characterized as set-transitive groups and were fully classified in \cite{BP55}. In this paper, we establish a general method that allows one to classify the permutation groups with \(n+r\) set-orbits for a given \(r\), and apply it to integers \(2 \leq r \leq 15\) using the computer algebra system GAP.