Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2041 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
422 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)(8,22)(9,29)(10,20)(11,36)(12,18)(13,17)(14,21)(15,24)(16,32)(19,38)(23,55)(25,42)(26,43)(27,45)(28,44)(30,41)(31,48)(33,53)(34,66)(35,50)(37,47)(39,49)(40,65)(46,58)(51,59)(52,61)(54,57)(56,67)(60,68)(62,64)(63,69);(6,8,14,28)(7,15,21,45)(9,30,34,60)(10,27,13,24)(11,26,38,59)(12,25,35,61)(16,46,49,55)(17,44,20,22)(18,51,50,43)(19,52,36,42)(23,37,58,33)(29,64,66,54)(31,57,40,62)(32,56,39,63)(41,65,68,48)(47,69,53,67);(6,9,10,31,14,34,13,40)(7,16,17,47,21,49,20,53)(8,23,24,56,28,58,27,63)(11,37,35,32,38,33,12,39)(15,41,22,54,45,68,44,64)(18,48,19,29,50,65,36,66)(25,60,59,57,61,30,26,62)(42,46,43,67,52,55,51,69)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6406);