Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#21677 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle, prime radical | C23×C5 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,13)(10,11)(14,15)(16,18)(17,19)(20,21)(22,23,26,25,27,29,34,33)(24,28,32,31,30,35,37,36);(7,11)(12,13)(14,16,17,20)(15,18,19,21)(22,24)(23,28)(25,31)(26,32)(27,30)(29,35)(33,36)(34,37);(6,8,10,9)(7,12,11,13)(15,19)(18,21)(24,30)(28,35)(31,36)(32,37);(6,9,10,8)(7,13,11,12)(22,25,26,29,27,33,34,23)(24,31,32,35,30,36,37,28)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14665);