Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#7476 |
| 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 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×C8 ◁ C22×C5×C16 ◁ G | |
162 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)(6,7,10,17,14,20,13,39)(8,21,23,53,26,55,32,54)(9,18,28,48,31,51,30,65)(11,34,33,50,29,16,36,47)(12,37,27,49,35,46,38,19)(15,40,41,22,45,57,44,59)(24,62,60,43,58,52,66,68)(25,64,61,42,63,67,56,69);(6,8,11,24,35,63,28,40,10,32,33,66,38,61,31,59,14,26,29,58,12,25,30,57,13,23,36,60,27,56,9,22)(7,15,18,42,49,68,47,54,39,41,65,67,37,62,16,21,20,45,51,69,19,43,50,53,17,44,48,64,46,52,34,55)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 464);