Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#38508 |
| 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 | |
78137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(6,7)(8,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,25)(23,28,24,27);(5,6,9,7)(8,11,12,10)(13,14,16,17)(15,19,20,18)(21,23,26,24)(22,27,25,28);(14,17)(15,20)(22,25)(23,24);(3,4);(1,2)(3,4);(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,17)(18,20)(21,24,26,23)(22,27,25,28);(5,8)(6,10)(7,11)(9,12)(13,15,16,20)(14,18,17,19)(21,25,26,22)(23,28,24,27)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((6,7)(8,12)(13,14)(15,19)(16,17)(18,20)(21,22,26,25)(23,28,24,27), (5,6,9,7)(8,11,12,10)(13,14,16,17)(15,19,20,18)(21,23,26,24)(22,27,25,28), (14,17)(15,20)(22,25)(23,24), (3,4), (1,2)(3,4), (5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,17)(18,20)(21,24,26,23)(22,27,25,28), (5,8)(6,10)(7,11)(9,12)(13,15,16,20)(14,18,17,19)(21,25,26,22)(23,28,24,27));