Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C32 · Schur multiplier (H2) = C2 |
| direct product | C5×C4⋊C32 |
| unambiguous semidirect product | C4⋊(C5×C32) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C16 | C22 |
| commutator | C2 | C2×C5×C32 |
| Frattini, cosocle | C2×C16 | C22×C5 |
| socle, prime radical | C22×C5 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C16 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,12,16,13,17,20,26,14,18,21,27,23,29,32,36,15,19,22,28,24,30,33,37,25,31,34,38,35,39,40,41);(6,7,8,9)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 554);