Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13144 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(11,16)(12,21)(13,17)(14,25)(18,33)(19,36)(20,30)(22,24)(23,31)(26,32)(27,34)(28,29)(35,39)(37,38);(6,8,9,7)(10,11,17,29,14,19,26,30,15,20,32,36,25,28,13,16)(12,18,31,40,23,33,21,34,24,35,38,41,37,39,22,27);(10,12,13,22,25,37,32,38,15,24,26,21,14,23,17,31)(11,18,16,27,28,39,36,41,20,35,30,34,19,33,29,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6132);