Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#21436 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
688 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)(8,11)(9,10)(12,13)(14,15)(16,19)(17,18)(20,21)(22,23,27,24,26,29,34,31)(25,28,33,30,32,35,37,36);(7,10)(11,13)(22,24,27,29,26,31,34,23)(25,30,33,35,32,36,37,28);(6,8,9,12)(7,11,10,13)(15,18)(19,21)(22,25,26,32)(23,28,29,35)(24,30,31,36)(27,33,34,37);(14,16,17,20)(15,19,18,21)(25,32)(28,35)(30,36)(33,37)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14424);