Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33850 |
| 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 | |
686 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)(10,11,17,13)(12,22,24,37)(14,28,31,20)(15,27,34,19)(16,29,33,21)(18,30,23,25)(26,40,35,38)(32,39,41,36);(10,12,17,24)(11,18,13,23)(14,30,31,25)(15,32,34,41)(16,35,33,26)(19,37,27,22)(20,38,28,40)(21,39,29,36);(7,9)(10,13,17,11)(12,23,24,18)(14,27,31,19)(15,28,34,20)(16,29,33,21)(22,25,37,30)(26,39,35,36)(32,40,41,38);(10,14)(11,19)(12,24)(13,27)(15,33)(16,34)(17,31)(18,23)(20,29)(21,28)(22,40)(25,41)(30,32)(37,38)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26838);