Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18524 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15)(9,13)(10,18)(11,14)(12,16)(17,20)(19,21)(22,23,27,30)(24,32,35,36)(25,28,26,31)(29,33,37,34);(6,8,11,17)(7,12,14,21)(9,16,10,19)(13,20,18,15)(23,28)(25,26)(29,36)(30,31)(32,37)(33,34);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,24,27,35)(23,29,30,37)(25,34,26,33)(28,36,31,32);(7,14)(12,21)(13,18)(15,20)(23,30)(28,31)(29,37)(32,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11530);