Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#40941 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
68993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22,25,28)(23,26,27,24);(5,7,8,12)(6,9,10,11)(13,14)(15,20)(16,17)(18,19)(21,23,25,27)(22,26,28,24);(7,12)(9,11)(14,17)(18,20)(21,23)(22,26)(24,28)(25,27);(3,4);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4);(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,17)(18,19)(21,24)(22,27)(23,28)(25,26)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((5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22,25,28)(23,26,27,24), (5,7,8,12)(6,9,10,11)(13,14)(15,20)(16,17)(18,19)(21,23,25,27)(22,26,28,24), (7,12)(9,11)(14,17)(18,20)(21,23)(22,26)(24,28)(25,27), (3,4), (5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,27)(26,28), (1,2)(3,4), (5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,17)(18,19)(21,24)(22,27)(23,28)(25,26));