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×#17837 |
| 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 | |
612 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)(6,7,12,9)(8,15,19,21)(10,14,11,20)(13,17,16,18)(22,23,26,30,27,31,25,29)(24,32,34,37,35,36,33,28);(6,8,10,17,12,19,11,18)(7,13,20,21,9,16,14,15)(22,24,26,34,27,35,25,33)(23,28,29,36,31,37,30,32);(6,9,12,7)(8,16,19,13)(10,14,11,20)(15,18,21,17)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32);(6,10,12,11)(7,14,9,20)(8,17,19,18)(13,15,16,21)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10825);