Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33393 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#102 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#578 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,3,4,5)(6,7,11,16,12,18,32,17)(8,19,22,41,23,46,31,45)(9,14,26,39,24,37,53,40)(10,27,29,15,30,42,58,43)(13,28,34,21,36,50,35,51)(20,44,49,63,47,65,57,66)(25,54,56,38,52,64,69,62)(33,55,60,48,59,68,61,67);(6,8,12,23)(7,13,18,36)(9,20,24,47)(10,28,30,50)(11,31,32,22)(14,33,37,59)(15,41,43,45)(16,35,17,34)(19,27,46,42)(21,29,51,58)(25,55,52,68)(26,57,53,49)(38,63,62,66)(39,61,40,60)(44,54,65,64)(48,56,67,69);(6,9)(7,14)(8,20)(10,25)(11,26)(12,24)(13,33)(15,38)(16,39)(17,40)(18,37)(19,44)(21,48)(22,49)(23,47)(27,54)(28,55)(29,56)(30,52)(31,57)(32,53)(34,60)(35,61)(36,59)(41,63)(42,64)(43,62)(45,66)(46,65)(50,68)(51,67)(58,69);(9,24)(14,37)(20,47)(25,52)(26,53)(33,59)(38,62)(39,40)(44,65)(48,67)(49,57)(54,64)(55,68)(56,69)(60,61)(63,66)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26381);