Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#23382 |
| 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 | |
606 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)(8,12)(9,10)(11,13)(14,15,17,19)(16,18,21,20)(22,23,26,29,25,27,32,34)(24,28,31,35,30,33,36,37);(6,7)(8,11)(9,10)(12,13)(14,15)(16,20)(17,19)(18,21)(23,27)(28,33)(29,34)(35,37);(7,10)(11,12)(14,16,17,21)(15,18,19,20)(22,24,25,30)(23,28,27,33)(26,31,32,36)(29,35,34,37);(6,8,9,13)(7,11,10,12)(15,19)(18,20)(23,27)(24,30)(29,34)(31,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16370);