Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#38109 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C25×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
74757 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)(7,9)(8,10)(11,12)(13,14,16,18)(15,19,20,17)(22,24)(23,27);(6,7)(10,11)(21,22,25,24)(23,26,27,28);(15,20)(17,19);(3,4);(5,7,9,6)(8,10,12,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,24,25,22)(23,28,27,26)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)(7,9)(8,10)(11,12)(13,14,16,18)(15,19,20,17)(22,24)(23,27), (6,7)(10,11)(21,22,25,24)(23,26,27,28), (15,20)(17,19), (3,4), (5,7,9,6)(8,10,12,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,26)(24,28)(25,27), (1,2)(3,4), (5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,24,25,22)(23,28,27,26));