Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#2986 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C24 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C22×C5 | C23×C4 |
| prime radical | C23×C5 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
330 conjugacy classes of subgroups; only the 256 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 256 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,25,28,26,29,32,35)(24,27,31,34,30,33,36,37);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(24,30)(27,33)(31,36)(34,37);(7,13)(11,14)(12,21)(19,20)(24,30)(27,33)(31,36)(34,37)250 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18663);