Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#43217 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C22×#2408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
68297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,9,7)(8,10,12,11)(13,14)(15,16)(17,18)(19,20)(21,22,25,27)(23,24,28,26);(6,7)(10,11)(14,18)(15,19)(21,23,25,28)(22,26,27,24);(6,7)(8,12)(22,27)(24,26);(3,4);(5,7,9,6)(8,10,12,11)(13,15,17,19)(14,16,18,20)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,20)(18,19)(21,24,25,26)(22,28,27,23)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((5,6,9,7)(8,10,12,11)(13,14)(15,16)(17,18)(19,20)(21,22,25,27)(23,24,28,26), (6,7)(10,11)(14,18)(15,19)(21,23,25,28)(22,26,27,24), (6,7)(8,12)(22,27)(24,26), (3,4), (5,7,9,6)(8,10,12,11)(13,15,17,19)(14,16,18,20)(21,23)(22,26)(24,27)(25,28), (1,2)(3,4), (5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,20)(18,19)(21,24,25,26)(22,28,27,23));