Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#2957 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×D4 |
| commutator | C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C2×C5 | C2×C4×D4 |
| prime radical | C2×C5×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×M16 ◁ G | |
452 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)(6,7,11,16,12,15,10,24)(8,17,19,13,20,28,25,27)(9,14,23,29,21,30,22,34)(18,31,33,26,32,36,35,37);(6,8)(7,13)(9,18)(10,25)(11,19)(12,20)(14,26)(15,27)(16,28)(17,24)(21,32)(22,35)(23,33)(29,36)(30,37)(31,34);(6,9,11,23,12,21,10,22)(7,14,16,29,15,30,24,34)(8,18,19,33,20,32,25,35)(13,26,28,36,27,37,17,31);(7,15)(9,21)(13,27)(16,24)(17,28)(18,32)(22,23)(33,35)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18634);