Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13122 |
| 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×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ G | |
326 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)(7,8)(10,11,16,21,17,20,35,38,14,19,23,24,15,32,34,40)(12,22,27,39,28,33,37,41,25,13,29,31,26,30,18,36);(6,7,9,8)(10,12,23,29,17,28,34,18,14,25,16,27,15,26,35,37)(11,13,24,39,20,30,40,41,19,22,21,31,32,33,38,36);(6,8,9,7)(10,13,14,22)(11,18,19,37)(12,24,25,21)(15,33,17,30)(16,31,23,39)(20,27,32,29)(26,38,28,40)(34,41,35,36)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6110);