Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#59091 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
98857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 24 points:
(11,15)(13,16)(17,18,22,20)(19,21,24,23);(6,8)(9,10,12,14)(11,13,15,16);(6,8)(17,19,22,24)(18,23,20,21);(3,4);(9,11,12,15)(10,13,14,16)(17,20,22,18)(19,21,24,23);(1,2)(3,4);(5,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,21)(18,19)(20,24)(22,23)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((11,15)(13,16)(17,18,22,20)(19,21,24,23), (6,8)(9,10,12,14)(11,13,15,16), (6,8)(17,19,22,24)(18,23,20,21), (3,4), (9,11,12,15)(10,13,14,16)(17,20,22,18)(19,21,24,23), (1,2)(3,4), (5,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,21)(18,19)(20,24)(22,23));