Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33842 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3283 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
664 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)(10,11,16,13)(12,20,23,39)(14,27,31,38)(15,26,29,19)(17,34,24,22)(18,33,25,30)(21,40,32,36)(28,37,41,35);(10,12,16,23)(11,17,13,24)(14,28,31,41)(15,32,29,21)(18,36,25,40)(19,37,26,35)(20,38,39,27)(22,33,34,30);(7,9)(10,13,16,11)(12,17,23,24)(14,25,31,18)(15,26,29,19)(20,34,39,22)(21,35,32,37)(27,30,38,33)(28,36,41,40);(10,14)(11,18)(12,21)(13,25)(15,33)(16,31)(17,35)(19,38)(20,39)(22,34)(23,32)(24,37)(26,27)(29,30)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26830);