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×#33464 |
| 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 | C5×#3414 | 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)(6,7,11,17,13,18,12,33)(8,19,22,45,23,46,34,42)(9,15,26,41,24,39,27,56)(10,28,30,43,32,58,31,16)(14,29,36,50,38,59,37,21)(20,44,49,66,47,65,57,64)(25,52,54,63,51,68,55,40)(35,53,61,67,60,69,62,48);(6,8)(7,14)(9,20)(10,29)(11,22)(12,34)(13,23)(15,35)(16,42)(17,36)(18,38)(19,28)(21,31)(24,47)(25,53)(26,49)(27,57)(30,50)(32,59)(33,37)(39,60)(40,64)(41,61)(43,45)(44,52)(46,58)(48,55)(51,69)(54,67)(56,62)(63,66)(65,68);(6,9,11,26,13,24,12,27)(7,15,17,41,18,39,33,56)(8,20,22,49,23,47,34,57)(10,25,30,54,32,51,31,55)(14,35,36,61,38,60,37,62)(16,40,28,52,43,63,58,68)(19,44,45,66,46,65,42,64)(21,48,29,53,50,67,59,69);(9,24)(15,39)(20,47)(25,51)(26,27)(35,60)(40,63)(41,56)(44,65)(48,67)(49,57)(52,68)(53,69)(54,55)(61,62)(64,66)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26452);