Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#42942 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
99161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 24 points:
(6,7)(9,10)(11,14)(12,13)(15,16)(19,23)(22,24);(6,7)(10,13)(14,15)(18,21)(22,24);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,23);(5,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,19,20,23)(18,22,21,24);(9,11,12,16)(10,14,13,15);(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)(11,14)(12,13)(15,16)(19,23)(22,24), (6,7)(10,13)(14,15)(18,21)(22,24), (5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,21)(22,23), (5,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,19,20,23)(18,22,21,24), (9,11,12,16)(10,14,13,15), (3,4), (1,2)(3,4));