Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C2×C42 |
| direct product | C5×#1744 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C5 | C22 |
| commutator | C2 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle, prime radical | C23×C5 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C5 ◁ G | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(10,11)(12,15)(13,14)(16,17)(18,19,20,22,21,23,24,25);(6,7,8,9)(11,14)(15,17);(10,12,13,16)(11,15,14,17)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5334);