Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#3414 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C5 |
| socle | C2×C5 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×C8 ◁ G | |
364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17)(9,15)(10,22)(11,26)(12,14)(13,19)(16,24)(18,34)(20,33)(21,31)(23,29)(25,36)(27,32)(28,35)(30,37);(6,8)(7,13)(9,18)(10,23)(11,27)(12,20)(14,28)(15,29)(16,32)(17,22)(19,24)(21,35)(25,37)(26,30)(31,34)(33,36);(7,14)(13,28)(15,26)(16,31)(17,33)(22,36)(29,30)(32,34);(6,9,12,11)(7,15,14,26)(8,18,20,27)(10,21,25,24)(13,29,28,30)(16,22,31,36)(17,34,33,32)(19,23,35,37)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19091);