Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18438 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
908 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)(7,11)(9,16)(12,18)(13,17)(15,20)(19,21)(22,23,27,31)(24,32,34,37)(25,30,26,29)(28,35,36,33);(6,7,10,13)(8,12,14,19)(9,17,16,11)(15,21,20,18)(22,24,27,34)(23,28,31,36)(25,33,26,35)(29,32,30,37);(6,8)(7,12)(9,15)(10,14)(11,18)(13,19)(16,20)(17,21)(25,26)(29,30)(32,37)(33,35);(8,14)(12,19)(15,20)(18,21)(22,25,27,26)(23,29,31,30)(24,33,34,35)(28,32,36,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11426);