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) = C25 |
| direct product | C2×C5×#1380 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | D8 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4)(5,6)(7,8,9,10)(12,17)(13,21)(14,23)(15,22)(18,19)(20,26)(24,25);(4,6)(11,12,14,18,15,19,13,17,16,20,21,25,22,24,23,26)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5543);