Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12829 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #418 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1804 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7)(8,18,21,13)(9,23)(10,27)(11,33)(12,15)(14,25)(16,31)(17,30)(19,46,47,65)(20,50,28,61)(22,37,53,45)(24,56)(26,59)(29,43)(32,60)(34,42)(35,48,44,58)(36,54,41,63)(38,64)(39,57)(40,51)(49,67,55,69)(52,66,62,68);(6,8,11,22,12,21,30,53)(7,13,33,45,15,18,17,37)(9,19,34,55,24,47,26,49)(10,28,31,63,32,20,51,54)(14,35,38,66,39,44,43,68)(16,41,60,61,40,36,27,50)(23,46,42,69,56,65,59,67)(25,58,64,52,57,48,29,62);(6,9)(7,14)(8,19)(10,29)(11,34)(12,24)(13,35)(15,39)(16,42)(17,43)(18,44)(20,52)(21,47)(22,55)(23,27)(25,31)(26,30)(28,62)(32,64)(33,38)(36,67)(37,68)(40,59)(41,69)(45,66)(46,50)(48,54)(49,53)(51,57)(56,60)(58,63)(61,65)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5817);