Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#1426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #367 |
| commutator | C16 | C2×C4×C5 |
| Frattini, cosocle | C2×C16 | C22×C5 |
| prime radical | C22×C5 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ C2×C5×C16 ◁ G | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(7,12,14,25)(8,17)(9,21)(10,22)(13,28)(15,29,32,37)(16,35,33,27)(18,23)(19,31)(20,30)(24,36,34,26);(6,7,13,29,9,15,8,12,22,33,17,26,18,24,28,35,11,14,31,37,23,32,20,25,10,16,30,36,21,34,19,27)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 559);