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×#3564 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| prime radical | C5×#538 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ G | |
936 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)(6,7,11,15)(8,16,17,13)(9,20,23,14)(10,12,22,26)(18,28,30,29)(19,27,31,24)(21,34,35,25)(32,37,33,36);(7,12)(8,17)(9,21)(13,27)(15,26)(16,24)(18,32)(19,31)(23,35)(28,29)(30,33)(36,37);(6,8,10,19)(7,13,12,24)(9,18,21,33)(11,17,22,31)(14,28,25,36)(15,16,26,27)(20,29,34,37)(23,30,35,32);(6,9,22,35)(7,14,26,34)(8,18,31,32)(10,21,11,23)(12,25,15,20)(13,28,27,37)(16,29,24,36)(17,30,19,33);(7,12)(13,24)(14,25)(15,26)(16,27)(20,34)(28,36)(29,37)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21431);