Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#22245 |
| 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 | |
686 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)(26,27,32,28,31,34,30,36)(29,33,38,35,39,41,40,37);(10,12)(11,16)(13,22)(14,20)(15,21)(17,25)(18,19)(23,24)(26,28,32,34,31,36,30,27)(29,37,38,33,39,35,40,41);(6,7,8,9)(10,13,15,14)(11,17,18,23)(12,20,21,22)(16,24,19,25)(26,29,31,39)(27,33,34,41)(28,35,36,37)(30,40,32,38);(10,14,15,13)(11,17,18,23)(12,20,21,22)(16,25,19,24)(26,30,31,32)(27,28,34,36)(29,38,39,40)(33,37,41,35)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15233);