Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#20324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#555 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
570 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)(8,14,17,21)(10,19,20,13)(12,15,18,16)(22,23,27,30)(24,28,26,35)(25,32,34,36)(29,33,37,31);(6,8,11,17)(7,12,9,18)(10,15,20,16)(13,21,19,14)(22,24,27,26)(23,28,30,35)(25,33,34,31)(29,32,37,36);(6,9,11,7)(8,12,17,18)(10,19,20,13)(14,15,21,16)(22,25,27,34)(23,29,30,37)(24,31,26,33)(28,36,35,32);(8,15)(10,20)(12,21)(13,19)(14,18)(16,17)(23,30)(28,35)(29,37)(32,36)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13312);