Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#3566 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
954 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)(7,13)(8,18)(9,23)(10,26)(14,31)(15,21)(17,30)(19,29)(22,35)(24,32)(25,36)(27,37);(6,7)(8,19)(9,24)(10,16)(11,13)(12,17)(14,18)(15,25)(20,26)(21,23)(22,31)(27,33)(28,30)(29,35)(32,36)(34,37);(6,8,11,22)(7,14,13,29)(9,20,25,16)(10,21,26,32)(12,18,28,35)(15,27,24,37)(17,19,30,31)(23,34,36,33);(6,9)(7,15)(8,20)(10,19)(11,25)(12,23)(13,24)(14,27)(16,22)(17,32)(18,34)(21,30)(26,31)(28,36)(29,37)(33,35);(6,10,12,27)(7,16,17,33)(8,21,18,24)(9,19,23,14)(11,26,28,37)(13,20,30,34)(15,22,32,35)(25,31,36,29)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21433);