Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#53098 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
60549 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(15,19)(18,20)(21,22,24,26)(23,27,28,25);(6,8)(7,11)(14,17)(18,20)(23,28)(25,27);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,26)(25,28);(5,8,10,6)(7,12,11,9)(13,15)(14,18)(16,19)(17,20)(21,23,24,28)(22,25,26,27);(5,9,10,12)(6,11,8,7);(3,4);(1,2)(3,4)184 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)(15,19)(18,20)(21,22,24,26)(23,27,28,25), (6,8)(7,11)(14,17)(18,20)(23,28)(25,27), (5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,26)(25,28), (5,8,10,6)(7,12,11,9)(13,15)(14,18)(16,19)(17,20)(21,23,24,28)(22,25,26,27), (5,9,10,12)(6,11,8,7), (3,4), (1,2)(3,4));