Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13169 |
| 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 | C23×C5 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
286 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)(10,11,15,20,16,22,30,21)(12,17,14,29,24,19,31,33)(13,25,27,38,28,32,40,39)(18,34,35,41,37,23,36,26);(7,9)(10,12,15,14,16,24,30,31)(11,17,20,29,22,19,21,33)(13,23,27,26,28,34,40,41)(18,32,35,39,37,25,36,38);(10,13,24,34,15,27,31,41,16,28,12,23,30,40,14,26)(11,18,29,39,21,36,17,32,22,37,33,38,20,35,19,25)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6157);