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×#1381 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | D8 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C23×C5 | #107 |
| prime radical | C24×C5 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
574 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)(7,8,11,16,13,19,25,18)(9,20,22,34,24,35,36,29)(10,17,12,15,26,32,27,33)(14,23,28,37,31,38,30,21);(4,6)(7,9,12,23,11,22,26,37,13,24,27,38,25,36,10,21)(8,14,17,29,18,30,33,35,19,31,32,34,16,28,15,20)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5546);