Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#40536 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
84185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(5,6,9,7)(8,10,12,11)(13,14,17,16)(15,19,20,18)(22,24)(23,27);(8,12)(10,11)(14,16)(15,20)(21,22,25,24)(23,26,27,28);(6,7)(8,12)(14,16)(18,19)(22,24)(26,28);(3,4);(5,7,9,6)(8,10,12,11)(13,15,17,20)(14,18,16,19)(21,23,25,27)(22,26,24,28);(1,2)(3,4);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,19,20,18)(21,24,25,22)(23,28,27,26)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((5,6,9,7)(8,10,12,11)(13,14,17,16)(15,19,20,18)(22,24)(23,27), (8,12)(10,11)(14,16)(15,20)(21,22,25,24)(23,26,27,28), (6,7)(8,12)(14,16)(18,19)(22,24)(26,28), (3,4), (5,7,9,6)(8,10,12,11)(13,15,17,20)(14,18,16,19)(21,23,25,27)(22,26,24,28), (1,2)(3,4), (5,8,9,12)(6,10,7,11)(13,16,17,14)(15,19,20,18)(21,24,25,22)(23,28,27,26));