Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C29 |
| direct product | C22×C4×C5×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C5 | C22 |
| commutator | C2 | C24×C4×C5 |
| Frattini, cosocle | C22 | C25×C5 |
| socle, prime radical | C24×C5 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C5 ◁ G | |
1664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(17,18,19,20,21);(3,4);(5,6,7,8);(9,10,11,12)(13,16,15,14);(9,13,11,15)(10,14,12,16)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21290);