Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#13440 |
| 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 | |
398 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)(10,11,16,22)(12,23,27,34)(13,17,14,30)(15,32,29,21)(18,24,36,25)(19,31,20,28)(26,33,39,37)(35,41,38,40);(11,17)(12,24)(13,28)(14,31)(15,29)(19,21)(20,32)(22,30)(23,37)(25,27)(26,41)(33,34)(35,38)(39,40);(7,9)(10,12,13,24,29,39,31,41,16,27,14,25,15,26,28,40)(11,18,19,34,32,38,30,37,22,36,20,23,21,35,17,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6428);