Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18692 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
924 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)(8,15)(9,13)(10,18)(11,14)(12,16)(17,20)(19,21)(22,23,27,31)(24,32,35,36)(25,30,26,29)(28,33,37,34);(6,8,11,17)(7,12,14,21)(9,16,10,19)(13,20,18,15)(22,24,27,35)(23,28,31,37)(25,33,26,34)(29,36,30,32);(6,7)(8,12)(9,18)(10,13)(11,14)(15,16)(17,21)(19,20)(23,29)(25,26)(28,36)(30,31)(32,37)(33,34);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,25,27,26)(23,30,31,29)(24,33,35,34)(28,32,37,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11680);