Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#2082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C5 |
| prime radical | C23×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,16,12,17,10,15)(8,18,21,13,22,29,20,30)(9,23,25,31,26,32,27,14)(19,34,35,37,24,28,36,33);(6,8,12,22)(7,13,17,30)(9,24,26,19)(10,21,11,20)(14,28,31,34)(15,29,16,18)(23,37,32,33)(25,35,27,36);(6,9,10,25,12,26,11,27)(7,14,15,23,17,31,16,32)(8,19,20,35,22,24,21,36)(13,28,18,33,30,34,29,37)115 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5672);