Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#3069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C5×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,12,17)(8,18,20,29)(9,23,25,31)(10,16,11,15)(13,22,28,21)(14,26,30,24)(19,33,34,27)(32,36,37,35);(6,8,10,21,12,20,11,22)(7,13,15,18,17,28,16,29)(9,19,24,35,25,34,26,36)(14,27,23,32,30,33,31,37);(6,9)(7,14)(8,19)(10,24)(11,26)(12,25)(13,27)(15,23)(16,31)(17,30)(18,32)(20,34)(21,35)(22,36)(28,33)(29,37);(8,20)(13,28)(18,29)(19,34)(21,22)(27,33)(32,37)(35,36)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18746);