Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C22×#46328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
90969 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)(11,15)(14,16)(17,18,21,19)(20,22,24,23);(6,7)(10,13)(14,16)(18,19)(22,23);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,22,24,23);(6,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24);(5,7,8,6);(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((5,6)(7,8)(11,15)(14,16)(17,18,21,19)(20,22,24,23), (6,7)(10,13)(14,16)(18,19)(22,23), (5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,22,24,23), (6,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24), (5,7,8,6), (3,4), (1,2)(3,4));