Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#3584 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C25×C5 |
| socle | C2×C5 | C23×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×#41 ◁ G | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17)(9,14)(10,15)(11,24)(12,16)(13,19)(18,31)(20,27)(21,29)(22,30)(23,34)(25,28)(26,33)(32,37)(35,36);(6,8)(7,13)(9,18)(10,19)(11,25)(12,20)(14,26)(15,27)(16,28)(17,24)(21,32)(22,33)(23,35)(29,36)(30,37)(31,34);(6,9)(7,14)(8,18)(10,22)(11,23)(12,21)(13,26)(15,30)(16,29)(17,31)(19,33)(20,32)(24,34)(25,35)(27,37)(28,36);(9,21)(14,29)(18,32)(22,23)(26,36)(30,34)(31,37)(33,35);(6,10,12,11)(7,15,16,24)(8,19,20,25)(9,22,21,23)(13,27,28,17)(14,30,29,34)(18,33,32,35)(26,37,36,31)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21451);