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×#3563 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C5×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15)(9,12)(10,18)(11,14)(13,19)(16,20)(17,21)(22,23,28,31)(24,26,29,25)(27,35,36,30)(32,33,37,34);(6,8,11,9)(7,12,14,15)(10,16,19,17)(13,20,18,21)(22,24,28,29)(23,25,31,26)(27,32,36,37)(30,33,35,34);(6,9,11,8)(7,12,14,15)(10,17,19,16)(13,20,18,21)(22,25,28,26)(23,29,31,24)(27,33,36,34)(30,37,35,32);(6,7)(8,15)(9,12)(10,13)(11,14)(16,21)(17,20)(18,19)(22,26,28,25)(23,24,31,29)(27,34,36,33)(30,32,35,37);(6,10)(7,13)(8,16)(9,17)(11,19)(12,20)(14,18)(15,21)(22,27)(23,30)(24,32)(25,33)(26,34)(28,36)(29,37)(31,35)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21430);