Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13320 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
540 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)(11,17)(12,22)(13,25)(14,15)(18,34)(19,26)(20,32)(21,33)(23,27)(24,30)(28,29)(31,36)(35,38)(37,40);(6,7)(8,9)(10,11,16,21)(12,18,24,37)(13,26,29,32)(14,20,15,19)(17,28,33,25)(22,38,30,36)(23,40,27,34)(31,39,35,41);(7,9)(10,12)(11,18)(13,27)(14,30)(15,22)(16,24)(17,31)(19,38)(20,36)(21,37)(23,29)(25,41)(26,34)(28,39)(32,40)(33,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6308);