Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C23×C5×D8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | D4 |
| commutator, Frattini, cosocle | C4 | C25×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C23×C4×C5 ◁ G | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(15,16,17,18,19);(3,4);(5,6);(7,8,9,10,11,12,13,14);(8,14)(9,13)(10,12)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21441);