Does there exist some sort of classification of incompressible groups?
Solution 1:
These were fully classified by Johnson in the paper 'Minimal Permutation Representations of Finite Groups'.
A group is incompressible iff it is isomorphic to one of the following:
- Cyclic group of prime power order $C_{p^n}$
- Generalised quaternion $2$-group $\langle x,y|x^{2^n}=1,x^{2^{n-1}}=y^2,x^y=x^{-1}\rangle$
- the Klein four-group $C_2\times C_2$
The proof is reasonably short so well worth looking up!
Reference: Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. MR 316540 DOI: 10.2307/2373739.