Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12642 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×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×C4×C5 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
430 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)(10,11,14,21,17,22,16,32)(12,23,26,36,29,41,28,34)(13,19,30,38,31,39,15,20)(18,33,35,24,37,27,40,25);(7,9)(10,12,15,27,16,28,31,24,17,29,30,33,14,26,13,25)(11,18,19,36,21,35,38,41,22,37,39,34,32,40,20,23);(10,13,14,30,17,31,16,15)(11,19,21,38,22,39,32,20)(12,24,26,27,29,25,28,33)(18,34,35,23,37,36,40,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5630);