Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#59372 |
| 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 | |
68081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(6,9)(7,11)(13,14,18,17)(15,20,16,19)(21,22,25,27)(23,26,28,24);(5,6,8,9)(7,10,11,12)(21,22)(23,24)(25,27)(26,28);(6,9)(10,12)(13,15,18,16)(14,19,17,20)(21,23,25,28)(22,24,27,26);(3,4);(5,7)(6,10)(8,11)(9,12)(13,16,18,15)(14,20,17,19)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4);(5,6)(7,10)(8,9)(11,12)(13,17,18,14)(15,20,16,19)(21,24,25,26)(22,23,27,28)256 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((6,9)(7,11)(13,14,18,17)(15,20,16,19)(21,22,25,27)(23,26,28,24), (5,6,8,9)(7,10,11,12)(21,22)(23,24)(25,27)(26,28), (6,9)(10,12)(13,15,18,16)(14,19,17,20)(21,23,25,28)(22,24,27,26), (3,4), (5,7)(6,10)(8,11)(9,12)(13,16,18,15)(14,20,17,19)(21,23)(22,26)(24,27)(25,28), (1,2)(3,4), (5,6)(7,10)(8,9)(11,12)(13,17,18,14)(15,20,16,19)(21,24,25,26)(22,23,27,28));