Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33465 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×C4×D8 |
| prime radical | C2×C5×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×M16 ◁ C5×#405 ◁ G | |
642 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)(6,7,11,18,13,16,12,33)(8,19,22,46,23,45,34,43)(9,15,26,40,24,41,27,56)(10,28,30,42,32,58,31,17)(14,29,37,50,36,59,38,21)(20,44,49,65,47,66,57,63)(25,52,54,64,51,68,55,39)(35,53,60,67,62,69,61,48);(6,8)(7,14)(9,20)(10,29)(11,22)(12,34)(13,23)(15,35)(16,36)(17,43)(18,37)(19,28)(21,31)(24,47)(25,53)(26,49)(27,57)(30,50)(32,59)(33,38)(39,63)(40,60)(41,62)(42,46)(44,52)(45,58)(48,55)(51,69)(54,67)(56,61)(64,65)(66,68);(6,9,11,26,13,24,12,27)(7,15,18,40,16,41,33,56)(8,20,22,49,23,47,34,57)(10,25,30,54,32,51,31,55)(14,35,37,60,36,62,38,61)(17,39,28,52,42,64,58,68)(19,44,46,65,45,66,43,63)(21,48,29,53,50,67,59,69);(7,16)(9,24)(14,36)(17,42)(18,33)(19,45)(20,47)(25,51)(26,27)(28,58)(37,38)(43,46)(48,67)(49,57)(53,69)(54,55)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26453);