Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
326 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)(10,11,16,33,17,23,31,22)(12,24,29,36,30,40,41,38)(13,19,25,20,14,21,15,32)(18,34,39,26,35,28,37,27);(6,7)(8,9)(10,12,25,34,16,29,14,26,17,30,15,28,31,41,13,27)(11,18,19,38,22,37,32,40,23,35,21,36,33,39,20,24);(7,9)(11,19)(12,26)(13,25)(14,15)(16,31)(18,35)(20,22)(21,23)(24,36)(27,30)(28,41)(29,34)(32,33)(38,40)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6458);