Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C22×#495725 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×#442 |
| commutator | C2×C8 | C25×C4 |
| Frattini, cosocle | C22×C8 | C26 |
| prime radical | C22×#33517 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ C24×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
95936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2);(3,4);(10,14,16,21)(11,17)(12,19)(15,20,23,22)(18,24);(5,6)(7,8);(10,15,16,23)(11,18)(14,20,21,22)(17,24);(6,8)(9,10,11,14,12,15,17,22,13,16,18,21,19,23,24,20)272 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2), (3,4), (10,14,16,21)(11,17)(12,19)(15,20,23,22)(18,24), (5,6)(7,8), (10,15,16,23)(11,18)(14,20,21,22)(17,24), (6,8)(9,10,11,14,12,15,17,22,13,16,18,21,19,23,24,20));