Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17834 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C22×C4⋊C4 |
| prime radical | C2×C5×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C4×C5×#41 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,11)(9,16)(12,17)(13,18)(15,20)(19,21)(22,23,26,29,27,30,25,35)(24,31,33,36,34,37,32,28);(6,7,9,11,10,13,16,18)(8,12,15,17,14,19,20,21)(22,24,25,32,27,34,26,33)(23,28,29,31,30,36,35,37);(6,8)(7,12)(9,15)(10,14)(11,17)(13,19)(16,20)(18,21);(8,14)(12,19)(15,20)(17,21)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10822);