Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#1881 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #120 |
| commutator | C2×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | #386 |
| prime radical | C5×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 108 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)(8,17,20,28)(9,21)(10,15)(11,25)(12,16)(13,19,29,24)(14,22)(18,33,36,27)(23,30)(26,31)(32,34,37,35);(6,8,10,24,12,20,11,19)(7,13,15,28,16,29,25,17)(9,18,22,35,23,36,26,34)(14,27,30,37,31,33,21,32);(6,9)(7,14)(8,18)(10,22)(11,26)(12,23)(13,27)(15,30)(16,31)(17,32)(19,34)(20,36)(21,25)(24,35)(28,37)(29,33)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5471);