Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C217×C4 |
| direct product | C22×#55312 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C25 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
78733 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)(14,17)(15,19)(21,22,25,26)(23,27,28,24);(5,6,8,9)(7,12,11,10)(13,14,16,17)(15,18,19,20);(6,9)(7,11)(14,17)(18,20)(21,23,25,28)(22,24,26,27);(3,4);(5,6)(7,12)(8,9)(10,11)(13,15,16,19)(14,18,17,20)(21,23)(22,24)(25,28)(26,27);(1,2)(3,4);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,17)(18,19)(21,24)(22,23)(25,27)(26,28)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((7,11)(10,12)(14,17)(15,19)(21,22,25,26)(23,27,28,24), (5,6,8,9)(7,12,11,10)(13,14,16,17)(15,18,19,20), (6,9)(7,11)(14,17)(18,20)(21,23,25,28)(22,24,26,27), (3,4), (5,6)(7,12)(8,9)(10,11)(13,15,16,19)(14,18,17,20)(21,23)(22,24)(25,28)(26,27), (1,2)(3,4), (5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,17)(18,19)(21,24)(22,23)(25,27)(26,28));