Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12648 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #442 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
422 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)(10,11,14,19)(12,22)(13,28,15,31)(16,33,29,21)(17,25)(18,32,20,30)(23,38)(24,37)(26,41)(27,35)(34,39)(36,40);(7,9)(10,12,15,25,16,26,32,39,14,27,13,24,29,23,30,40)(11,17,20,35,21,36,31,41,19,37,18,22,33,34,28,38);(12,23,27,26)(13,15)(17,34,37,36)(18,20)(22,38,35,41)(24,40,25,39)(28,31)(30,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5636);