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) = C2×C4 |
| direct product | C5×#12597 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #117 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1746 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C5×#405 ◁ G | |
258 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)(6,7,10,15,11,17,27,39,12,16,28,38,29,40,54,64)(8,18,21,42,22,43,47,66,20,13,30,32,46,33,56,59)(9,14,24,35,26,36,52,62,25,37,51,63,53,57,68,69)(19,41,45,65,23,48,50,67,44,31,55,58,49,34,60,61);(6,8)(7,13,16,18)(9,23,25,49)(10,21)(11,22)(12,20)(14,34)(15,32,38,42)(17,33,40,43)(19,26,44,53)(24,50,51,60)(27,47)(28,30)(29,46)(31,57)(35,61)(36,41)(37,48)(39,59,64,66)(45,52,55,68)(54,56)(58,69)(62,65)(63,67);(6,9)(7,14)(8,19,20,44)(10,24)(11,26)(12,25)(13,31,18,41)(15,35)(16,37)(17,36)(21,45,30,55)(22,23,46,49)(27,52)(28,51)(29,53)(32,58,42,65)(33,34,43,48)(38,63)(39,62)(40,57)(47,50,56,60)(54,68)(59,61,66,67)(64,69)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5585);