Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#1382 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | D8 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C23×C5 | #102 |
| prime radical | C24×C5 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2)(39,40,41,42,43);(3,4,5,6)(7,8)(9,18)(10,23)(11,26)(12,28)(13,17)(14,19)(15,25)(16,24)(20,30)(21,38)(22,34)(27,31)(29,37)(32,36)(33,35);(7,9,11,20,24,35,25,36,13,22,10,19,12,21,27,37)(8,14,15,18,28,32,26,38,17,30,31,34,16,29,23,33)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5548);