Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C218 |
| direct product | C22×#48600 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
67989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 7 generators on 28 points:
(7,11)(10,12)(15,19)(17,20)(22,27)(24,25);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,28)(26,27);(6,9)(10,12)(13,14,16,18)(15,20,19,17)(21,23,26,28)(22,24,27,25);(3,4);(5,7,8,11)(6,10,9,12)(13,14)(15,20)(16,18)(17,19)(21,24,26,25)(22,23,27,28);(1,2)(3,4);(5,6)(7,12)(8,9)(10,11)(13,15,16,19)(14,17,18,20)(21,25,26,24)(22,28,27,23)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((7,11)(10,12)(15,19)(17,20)(22,27)(24,25), (5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,24)(25,28)(26,27), (6,9)(10,12)(13,14,16,18)(15,20,19,17)(21,23,26,28)(22,24,27,25), (3,4), (5,7,8,11)(6,10,9,12)(13,14)(15,20)(16,18)(17,19)(21,24,26,25)(22,23,27,28), (1,2)(3,4), (5,6)(7,12)(8,9)(10,11)(13,15,16,19)(14,17,18,20)(21,25,26,24)(22,28,27,23));