Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#2247 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C4×C5×C8 ◁ G | |
176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11,15,19)(12,20,24,36)(13,25,27,40)(14,28,26,18)(16,29,31,22)(17,32,33,21)(23,39,38,35)(30,37,34,41);(10,12,21,37,14,23,27,29,15,24,32,41,26,38,13,22)(11,16,25,35,18,30,33,36,19,31,40,39,28,34,17,20);(6,7,8,9)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5837);