Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7863 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C42 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | C2×#95 |
| prime radical | C24×C5 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15,17,21)(9,13)(10,19)(11,14)(12,16,20,18)(22,23,25,28,26,29,32,35)(24,27,31,34,30,33,36,37);(6,8,10,18,11,17,9,16)(7,12,13,15,14,20,19,21)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(24,30)(27,33)(31,36)(34,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 851);