Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#22255 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ 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)(6,7,9,8)(10,11)(12,19)(13,17)(14,23)(15,18)(16,21)(20,25)(22,24)(28,34)(29,36)(31,37)(33,39)(35,40)(38,41);(6,8,9,7)(10,12)(11,16)(13,22)(14,20)(15,21)(17,25)(18,19)(23,24)(26,27)(28,31)(29,33)(30,32)(34,37)(35,38)(36,39)(40,41);(10,13,15,14)(11,17,18,23)(12,20,21,22)(16,24,19,25)(26,28,29,34,30,35,36,40)(27,31,33,37,32,38,39,41);(10,14,15,13)(11,17,18,23)(12,20,21,22)(16,25,19,24)(27,32)(28,35)(33,39)(34,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15243);