Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#2223 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C5 |
| prime radical | C5×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#131 ◁ G | |
144 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 60 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)(7,13)(8,19)(9,22)(10,11)(14,23)(15,30)(16,27)(17,26)(18,28)(20,35)(21,24)(25,31)(32,37)(33,36);(6,7,14,26,11,17,9,13,12,18,22,27,10,16,23,28)(8,15,29,37,19,31,20,25,21,32,34,30,24,33,35,36);(6,8)(7,15)(9,20)(10,24)(11,19)(12,21)(13,25)(14,29)(16,33)(17,31)(18,32)(22,34)(23,35)(26,37)(27,30)(28,36)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5813);