Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13134 |
| 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)(7,8)(10,11,16,21,14,20,17,32)(12,22,25,39,24,18,15,30)(13,26,29,38,27,41,33,34)(19,31,36,23,35,40,37,28);(6,7,9,8)(10,12,17,15,14,24,16,25)(11,18,32,39,20,22,21,30)(13,23,33,31,27,28,29,40)(19,34,37,41,35,38,36,26);(6,8,9,7)(10,13,14,27)(11,19,20,35)(12,23,24,28)(15,31,25,40)(16,29,17,33)(18,34,22,38)(21,36,32,37)(26,39,41,30)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6122);