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×#3568 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
958 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)(6,7,11,13)(8,17,21,19)(9,12,16,22)(10,24,25,15)(14,28,27,18)(20,33,34,31)(23,35,30,26)(29,32,36,37);(7,13)(8,18)(9,16)(14,19)(15,24)(17,27)(20,32)(21,28)(23,30)(29,33)(31,36)(34,37);(6,8,11,21)(7,14,13,27)(9,19,16,17)(10,20,25,34)(12,18,22,28)(15,29,24,36)(23,33,30,31)(26,32,35,37);(6,9,22,7)(8,19,28,14)(10,23,35,15)(11,16,12,13)(17,18,27,21)(20,33,37,29)(24,25,30,26)(31,32,36,34);(6,10,22,35)(7,15,9,23)(8,20,28,37)(11,25,12,26)(13,24,16,30)(14,29,19,33)(17,31,27,36)(18,32,21,34)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21435);