Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13152 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C4×D8 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
324 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)(10,11,16,22,17,24,35,23)(12,25,29,38,13,30,33,41)(14,34,36,20,26,21,15,37)(18,32,19,31,40,28,39,27);(10,12,26,31,35,33,36,32,17,13,14,27,16,29,15,28)(11,18,20,25,22,19,21,38,24,40,37,30,23,39,34,41);(7,9)(10,13,14,31,35,29,15,32,17,12,26,27,16,33,36,28)(11,19,20,38,22,40,21,30,24,39,37,41,23,18,34,25)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6140);