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×#33835 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
670 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)(12,20,24,39)(14,27,31,38)(15,26,29,19)(17,34,21,23)(18,33,25,30)(22,40,32,36)(28,37,41,35);(10,12,16,24)(11,17,13,21)(14,28,31,41)(15,32,29,22)(18,36,25,40)(19,37,26,35)(20,38,39,27)(23,33,34,30);(7,9)(10,13,16,11)(12,21,24,17)(14,25,31,18)(15,26,29,19)(20,23,39,34)(22,37,32,35)(27,30,38,33)(28,40,41,36);(10,14)(11,18)(12,22)(13,25)(15,33)(16,31)(17,35)(19,38)(20,39)(21,37)(23,34)(24,32)(26,27)(29,30)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26823);