Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13332 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
566 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)(12,22)(13,18)(14,20)(15,31)(16,21)(17,24)(19,28)(23,32)(25,37)(26,39)(27,38)(29,35)(30,36)(33,41)(34,40);(7,9)(10,12,16,26)(11,17,21,34)(13,27,30,41)(14,23,15,25)(18,22,36,39)(19,37,35,32)(20,38,31,33)(24,29,40,28);(11,18)(12,23)(13,28)(14,15)(17,32)(19,20)(21,36)(22,38)(24,40)(25,26)(29,30)(31,35)(33,39)(34,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6320);