Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#3293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ G | |
444 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 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)(6,7,13,18)(8,19,22,24)(9,21,14,28)(10,15,27,33)(11,17,12,16)(20,35,37,31)(23,36,30,25)(26,34,29,32);(6,8,13,22)(7,14,18,9)(10,25,27,36)(11,28,12,21)(15,31,33,35)(16,24,17,19)(20,29,37,26)(23,32,30,34);(6,9,13,14)(7,8,18,22)(10,23,27,30)(11,24,12,19)(15,20,33,37)(16,21,17,28)(25,34,36,32)(26,35,29,31);(6,10)(7,15)(8,20)(9,23)(11,29)(12,26)(13,27)(14,30)(16,34)(17,32)(18,33)(19,35)(21,36)(22,37)(24,31)(25,28)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18970);