Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#7475 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #367 |
| commutator | C16 | C42×C5 |
| Frattini, cosocle | C22×C16 | C22×C5 |
| socle | C2×C5 | #1380 |
| prime radical | C23×C5 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×C8 ◁ C22×C5×C16 ◁ G | |
170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 69 points:
(1,2,3,4,5)(6,7)(8,20,22,48)(9,16)(10,30)(11,33)(12,36)(13,19)(14,21,37,51)(15,27)(17,35)(18,34)(23,56,52,47)(24,58,54,69)(25,60,55,49)(26,43)(28,63)(29,46)(31,41)(32,42)(38,50,65,59)(39,67,66,57)(40,68,64,53)(44,62)(45,61);(6,8,10,23,31,54,61,68,34,55,62,67,35,59,29,51,13,22,27,52,11,24,28,53,12,25,32,57,26,50,9,21)(7,14,16,38,43,66,42,49,36,64,63,69,33,56,15,20,19,37,46,65,17,39,44,60,18,40,45,58,41,47,30,48)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 463);