Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#463 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C22×C5×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(15,16,17,18,19);(4,6)(7,8)(9,11)(10,12)(13,14);(3,4,5,6)(8,11)(12,13);(8,12)(10,14)(11,13)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18708);