Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#22243 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle, prime radical | C23×C5 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11)(12,19)(13,17)(14,23)(15,18)(16,21)(20,25)(22,24)(26,27,32,28,31,35,30,34)(29,33,37,36,38,40,39,41);(10,12)(11,16)(13,22)(14,20)(15,21)(17,25)(18,19)(23,24)(26,28,32,35,31,34,30,27)(29,36,37,40,38,41,39,33);(6,7,8,9)(10,13,15,14)(11,17,18,23)(12,20,21,22)(16,24,19,25)(26,29,31,38)(27,33,35,40)(28,36,34,41)(30,39,32,37);(10,14,15,13)(11,17,18,23)(12,20,21,22)(16,25,19,24)(26,30,31,32)(27,34,35,28)(29,37,38,39)(33,36,40,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15231);