Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19846 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#127 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,9)(8,14,16,21)(10,18,20,13)(12,19,17,15)(22,23,27,30)(24,28,26,34)(25,32,33,36)(29,35,37,31);(6,8,11,16)(7,12,9,17)(10,19,20,15)(13,21,18,14)(22,24,27,26)(23,28,30,34)(25,31,33,35)(29,36,37,32);(22,25,27,33)(23,29,30,37)(24,31,26,35)(28,36,34,32);(6,9,11,7)(8,14,16,21)(10,13,20,18)(12,19,17,15)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12834);