Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#2098 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | #399 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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)(6,7,10,15,11,16,12,26)(8,17,19,29,21,33,20,13)(9,22,24,32,25,14,27,31)(18,34,23,37,35,30,36,28);(6,8,12,19,11,21,10,20)(7,13,26,17,16,29,15,33)(9,23,27,35,25,36,24,18)(14,30,32,28,22,34,31,37);(6,9)(7,14)(8,18)(10,24)(11,25)(12,27)(13,28)(15,31)(16,22)(17,34)(19,23)(20,36)(21,35)(26,32)(29,37)(30,33)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5688);