Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#12902 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #127 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1843 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
278 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)(6,7,10,15,11,16,26,37,12,17,27,38,28,39,52,63)(8,13,20,31,21,32,45,57,19,30,43,55,44,56,65,68)(9,14,24,35,25,36,51,62,23,34,49,60,50,61,67,69)(18,40,42,64,22,46,48,66,41,29,53,54,47,33,58,59);(6,8)(7,13)(9,22,23,47)(10,20)(11,21)(12,19)(14,33,34,46)(15,31)(16,32)(17,30)(18,25,41,50)(24,48,49,58)(26,45)(27,43)(28,44)(29,36,40,61)(35,59,60,66)(37,57)(38,55)(39,56)(42,51,53,67)(52,65)(54,62,64,69)(63,68);(6,9)(7,14)(8,18,19,41)(10,24)(11,25)(12,23)(13,29,30,40)(15,35)(16,36)(17,34)(20,42,43,53)(21,22,44,47)(26,51)(27,49)(28,50)(31,54,55,64)(32,33,56,46)(37,62)(38,60)(39,61)(45,48,65,58)(52,67)(57,59,68,66)(63,69)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5890);