Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#47889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
72769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(6,8)(7,11)(13,14,16,19)(15,18,20,17)(21,22,25,26)(23,24,28,27);(5,6,9,8)(7,12,11,10)(22,26)(24,27);(6,8)(10,12)(13,15,16,20)(14,17,19,18)(21,23,25,28)(22,27,26,24);(3,4);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,27)(26,28);(1,2)(3,4);(5,8,9,6)(7,10,11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,27)(24,26)(25,28)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((6,8)(7,11)(13,14,16,19)(15,18,20,17)(21,22,25,26)(23,24,28,27), (5,6,9,8)(7,12,11,10)(22,26)(24,27), (6,8)(10,12)(13,15,16,20)(14,17,19,18)(21,23,25,28)(22,27,26,24), (3,4), (5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,27)(26,28), (1,2)(3,4), (5,8,9,6)(7,10,11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,27)(24,26)(25,28));