Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#1342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #327 |
| commutator | C2×C8 | C2×C4×C5 |
| Frattini, cosocle | C2×M16 | C22×C5 |
| prime radical | C23×C5 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×M16 ◁ G | |
100 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)(6,7,10,15,12,16,11,27)(8,17,19,31,21,34,20,37)(9,22,24,14,26,32,25,33)(13,23,28,36,30,35,29,18);(6,8,12,21)(7,13,16,30)(9,23,26,35)(10,20,11,19)(14,31,33,37)(15,28,27,29)(17,32,34,22)(18,25,36,24)100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 475);