Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#33438 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C23×C8 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,20,16,21,30,37,14,19,28,35,29,36,32,41)(12,22,24,38,25,26,34,40,23,17,31,33,39,13,18,27);(7,9)(10,12)(11,17)(13,21)(14,23)(15,24)(16,25)(18,32)(19,22)(20,33)(26,36)(27,37)(28,31)(29,39)(30,34)(35,38)(40,41);(10,13,15,27,16,22,30,38,14,26,28,40,29,17,32,33)(11,18,20,12,21,24,37,25,19,34,35,23,36,31,41,39);(12,23)(13,26)(17,22)(18,34)(24,31)(25,39)(27,40)(33,38)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26426);