Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12614 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C16 | D4 |
| commutator | C4 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C2×C5×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C16 ◁ C22×C5×C16 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,14,20,15,19,27,34,16,21,28,35,13,17,26,30)(12,18,23,32,24,31,37,40,25,33,38,41,22,29,36,39);(7,9)(11,17,21,19)(18,29,33,31)(20,30,35,34)(32,39,41,40);(10,12,14,23,15,24,27,37,16,25,28,38,13,22,26,36)(11,18,20,32,19,31,34,40,21,33,35,41,17,29,30,39)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5602);