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×#60295 |
| 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×#3444 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,13)(10,12)(11,15)(16,20)(17,19)(18,21)(22,23,25,29,26,28,32,35)(24,27,31,34,30,33,36,37);(6,8)(7,11)(9,16)(10,15)(12,14)(13,18)(17,21)(19,20)(22,24)(23,27)(25,31)(26,30)(28,33)(29,34)(32,36)(35,37);(8,15)(11,14)(16,21)(18,20)(23,28)(24,30)(29,35)(31,36);(7,12)(11,14)(13,19)(18,20)(24,30)(27,33)(31,36)(34,37);(6,9,10,17)(7,13,12,19)(8,16,15,21)(11,18,14,20)(24,30)(27,33)(31,36)(34,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53283);