Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#10868 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #382 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,21,24,17)(13,26)(14,30)(15,20)(16,19)(18,27)(22,36,38,40)(23,32,25,35)(28,33)(29,41)(31,37,34,39);(7,9)(10,12,15,25,16,24,14,23)(11,17,20,32,19,21,30,35)(13,22,28,39,29,38,27,37)(18,31,26,40,33,34,41,36);(10,13,14,27,16,29,15,28)(11,18,20,26,19,33,30,41)(12,22,23,37,24,38,25,39)(17,31,32,40,21,34,35,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3856);