Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#35930 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
97529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 24 points:
(5,6,7,8)(9,10,13,12)(11,14,16,15)(17,18)(19,20)(21,23)(22,24);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,24)(21,22);(5,6)(7,8)(10,12)(14,15)(19,21)(20,23);(5,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,23)(18,19,24,21);(9,12,13,10)(11,14,16,15)(17,21,22,19)(18,23,24,20);(3,4);(1,2)(3,4)256 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((5,6,7,8)(9,10,13,12)(11,14,16,15)(17,18)(19,20)(21,23)(22,24), (5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,24)(21,22), (5,6)(7,8)(10,12)(14,15)(19,21)(20,23), (5,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,23)(18,19,24,21), (9,12,13,10)(11,14,16,15)(17,21,22,19)(18,23,24,20), (3,4), (1,2)(3,4));