Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13196 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C2×C5×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ 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)(7,9)(10,11,16,22,17,21,32,38,14,20,24,19,15,33,13,30)(12,23,28,37,29,39,25,40,26,18,34,35,27,36,31,41);(6,7,8,9)(10,12,24,34,17,29,13,31,14,26,16,28,15,27,32,25)(11,18,19,37,21,36,30,40,20,23,22,35,33,39,38,41);(10,13,15,24,14,32,17,16)(11,19,21,30,20,22,33,38)(12,25,27,28,26,31,29,34)(18,35,36,41,23,37,39,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6184);