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×#22626 |
| 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 | |
744 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)(7,10)(11,12)(14,15)(16,18)(17,19)(20,21)(22,23,25,28,26,29,32,35)(24,27,31,34,30,33,36,37);(6,7,9,10)(8,12,13,11)(14,16,17,20)(15,18,19,21)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(15,19)(18,21)(24,30)(27,33)(31,36)(34,37);(6,8,9,13)(7,11,10,12)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15614);