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×#13145 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(11,16)(12,21)(13,25)(14,26)(17,27)(18,34)(19,32)(20,31)(22,24)(23,38)(28,35)(29,30)(33,37)(36,39);(6,8,9,7)(10,11,17,32,14,19,13,16,15,20,25,30,26,29,27,31)(12,18,33,41,23,34,22,28,24,36,38,40,37,39,21,35);(10,12,13,22,26,37,17,33,15,24,27,21,14,23,25,38)(11,18,16,28,29,39,32,41,20,36,31,35,19,34,30,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6133);