Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#11591 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C5 | D8 |
| commutator | C8 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C2×C5×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,8,10,9,11,12,13)(14,15)(16,25)(17,22)(18,30)(19,34)(20,24)(21,27)(23,32)(26,41)(28,42)(29,40)(31,36)(33,44)(35,39)(37,45)(38,43);(14,16)(15,21)(17,26)(18,31)(19,35)(20,28)(22,36)(23,39)(24,38)(25,30)(27,32)(29,43)(33,45)(34,37)(40,41)(42,44);(14,17,20,19)(15,22,24,34)(16,26,28,35)(18,29,33,32)(21,36,38,37)(23,30,40,44)(25,41,42,39)(27,31,43,45)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4579);