Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12641 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | D8 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C4×D8 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
438 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)(8,9)(10,11,14,21,17,22,16,31)(12,23,25,33,28,39,27,41)(13,19,29,36,30,37,15,20)(18,32,34,40,35,26,38,24);(7,9)(10,12,15,26,16,27,30,40,17,28,29,32,14,25,13,24)(11,18,19,33,21,34,36,39,22,35,37,41,31,38,20,23);(10,13,14,29,17,30,16,15)(11,19,21,36,22,37,31,20)(12,24,25,32,28,40,27,26)(18,33,34,39,35,41,38,23)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5629);