Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C217×C42 |
| direct product | C22×#37639 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
106381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24);(6,7)(10,14)(11,16)(18,22)(19,23);(5,6,8,7)(11,16)(12,15)(17,19)(18,20)(21,23)(22,24);(5,7,8,6)(9,11,13,16)(10,15,14,12)(17,18)(19,24)(20,23)(21,22);(9,12,13,15)(10,11,14,16)(17,20,21,24)(18,19,22,23);(3,4);(1,2)(3,4)208 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (6,7)(10,14)(11,16)(18,22)(19,23), (5,6,8,7)(11,16)(12,15)(17,19)(18,20)(21,23)(22,24), (5,7,8,6)(9,11,13,16)(10,15,14,12)(17,18)(19,24)(20,23)(21,22), (9,12,13,15)(10,11,14,16)(17,20,21,24)(18,19,22,23), (3,4), (1,2)(3,4));