Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13525 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #2092 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
294 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)(6,7,10,16,35,51,11,17,13,19,37,53,12,18,36,52)(8,20,22,42,24,41,23,44,26,54,38,43,25,49,33,14)(9,27,29,15,39,46,34,55,32,50,64,48,31,62,30,47)(21,45,57,56,28,63,61,69,60,68,65,67,59,40,58,66);(6,8,13,26)(7,14,19,44)(9,28)(10,33,37,23)(11,38,36,22)(12,24,35,25)(15,45)(16,49,53,41)(17,54,52,20)(18,42,51,43)(21,39)(27,56)(29,57)(30,61)(31,60)(32,59)(34,58)(40,55)(46,66)(47,63)(48,68)(50,67)(62,69)(64,65);(6,9)(7,15)(8,21)(10,34)(11,29)(12,39)(13,32)(14,40)(16,50)(17,46)(18,55)(19,48)(20,56)(22,61)(23,57)(24,59)(25,28)(26,60)(27,53)(30,37)(31,35)(33,65)(36,64)(38,58)(41,66)(42,68)(43,45)(44,63)(47,51)(49,69)(52,62)(54,67)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6513);