Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13034 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1879 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
462 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)(6,7,11,16,12,18,37,49,13,19,10,17,33,48,34,52)(8,20,23,41,24,46,35,42,25,56,22,53,38,50,31,14)(9,26,28,54,29,47,32,15,30,43,27,44,39,51,36,45)(21,40,58,64,59,65,61,68,60,66,57,69,63,67,62,55);(6,8)(7,14,19,42)(9,21)(10,31)(11,35)(12,38)(13,25)(15,40,45,66)(16,46,17,50)(18,53,48,41)(20,52,56,49)(22,34)(23,37)(24,33)(26,55,43,68)(27,62)(28,61)(29,63)(30,60)(32,58)(36,57)(39,59)(44,67,54,65)(47,69,51,64);(6,9,13,30)(7,15,19,45)(8,21,25,60)(10,32,11,36)(12,39,33,29)(14,40,42,66)(16,47,17,51)(18,54,48,44)(20,55,56,68)(22,61,23,62)(24,63,38,59)(26,52,43,49)(27,34,28,37)(31,58,35,57)(41,67,53,65)(46,69,50,64)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6022);