Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60337 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C5×#3482 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,26,25,27,29,34,33)(24,28,32,31,30,35,37,36);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24)(23,28)(25,31)(26,32)(27,30)(29,35)(33,36)(34,37);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(24,30)(28,35)(31,36)(32,37);(7,13)(11,14)(12,21)(19,20)(24,30)(28,35)(31,36)(32,37);(22,25,26,29,27,33,34,23)(24,31,32,35,30,36,37,28)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53325);