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) = C2×C4 |
| direct product | C4×C5×#368 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | D8 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#107 |
| prime radical | C23×C5 | C2×Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4)(37,38,39,40,41);(5,6,9,14,11,15,10,25)(7,16,19,31,21,32,20,35)(8,22,23,29,24,36,17,13)(12,26,27,34,28,33,30,18);(5,7,17,33,10,20,24,34,11,21,23,26,9,19,8,18)(6,12,22,31,14,27,29,32,15,28,36,35,25,30,13,16)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5663);