Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#1684 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
144 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,13)(8,14,17,21)(9,18,15,12)(11,19,20,16)(22,23)(24,30)(25,27)(26,34)(28,32)(29,33)(31,37)(35,36);(6,8,15,19,10,17,9,16)(7,11,18,21,13,20,12,14)(23,27)(28,36)(29,34)(30,37);(22,24,25,31)(23,28,27,36)(26,35,33,32)(29,37,34,30)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5274);