Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C42 · Schur multiplier (H2) = C220×C4 |
| direct product | C22×#513714 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C25 |
| commutator | C22 | C25×C42 |
| Frattini, cosocle | C24 | C27 |
| prime radical | C26×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1077016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(17,18,21,22)(19,20,23,24);(6,8)(9,10,11,12)(13,14,16,15)(18,22)(20,24);(6,8)(9,10,11,12)(17,18,21,22)(19,24,23,20);(3,4);(6,8)(9,10,11,12)(13,15,16,14)(17,19)(18,20)(21,23)(22,24);(1,2)(3,4);(5,6)(7,8)(9,10)(11,12)(17,20)(18,23)(19,22)(21,24)704 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((17,18,21,22)(19,20,23,24), (6,8)(9,10,11,12)(13,14,16,15)(18,22)(20,24), (6,8)(9,10,11,12)(17,18,21,22)(19,24,23,20), (3,4), (6,8)(9,10,11,12)(13,15,16,14)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4), (5,6)(7,8)(9,10)(11,12)(17,20)(18,23)(19,22)(21,24));