Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#3028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C5×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
716 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 212 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)(6,7,11,15)(8,12,18,25)(9,13,20,27)(10,21,22,14)(16,28,30,23)(17,31,32,24)(19,33,34,26)(29,35,37,36);(6,8)(7,12)(9,16)(10,17)(11,18)(13,23)(14,24)(15,25)(19,29)(20,30)(21,31)(22,32)(26,35)(27,28)(33,36)(34,37);(6,9)(7,13)(8,16)(10,19)(11,20)(12,23)(14,26)(15,27)(17,29)(18,30)(21,33)(22,34)(24,35)(25,28)(31,36)(32,37);(6,10,11,22)(7,14,15,21)(8,17,18,32)(9,19,20,34)(12,24,25,31)(13,26,27,33)(16,29,30,37)(23,35,28,36)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18705);