Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218×C4 |
| direct product | C22×#39881 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
94325 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)(9,10,13,12)(11,14,15,16)(17,18,20,22)(19,21,23,24);(5,6)(7,8)(11,15)(14,16)(17,18,20,22)(19,21,23,24);(6,8)(19,23)(21,24);(3,4);(5,6)(7,8)(9,11,13,15)(10,14,12,16)(17,18)(19,24)(20,22)(21,23);(1,2)(3,4);(5,6)(7,8)(9,12,13,10)(11,14,15,16)(17,19)(18,21)(20,23)(22,24)232 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)(11,14,15,16)(17,18,20,22)(19,21,23,24), (5,6)(7,8)(11,15)(14,16)(17,18,20,22)(19,21,23,24), (6,8)(19,23)(21,24), (3,4), (5,6)(7,8)(9,11,13,15)(10,14,12,16)(17,18)(19,24)(20,22)(21,23), (1,2)(3,4), (5,6)(7,8)(9,12,13,10)(11,14,15,16)(17,19)(18,21)(20,23)(22,24));