Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C4 · Schur multiplier (H2) = C220×C4 |
| direct product | C22×#519285 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C23 | C26×C4 |
| Frattini, cosocle | C24 | C27 |
| prime radical | C29 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2882696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 24 points:
(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,20,22)(19,23,24,21);(6,8)(10,12)(17,18,20,22)(19,21,24,23);(9,10)(11,12);(5,6)(7,8)(9,10)(11,12)(17,18)(19,21)(20,22)(23,24);(17,19)(18,21)(20,24)(22,23);(3,4);(1,2)(3,4)560 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((5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,20,22)(19,23,24,21), (6,8)(10,12)(17,18,20,22)(19,21,24,23), (9,10)(11,12), (5,6)(7,8)(9,10)(11,12)(17,18)(19,21)(20,22)(23,24), (17,19)(18,21)(20,24)(22,23), (3,4), (1,2)(3,4));