Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×#1328 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #99 |
| commutator | C23 | C2×C5×C8 |
| Frattini, cosocle | C2×M16 | C22×C5 |
| socle | C2×C5 | #324 |
| prime radical | C23×C5 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×M16 ◁ G | |
90 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)(6,7,9,14,10,15,20,28,11,13,19,25,21,26,32,35)(8,12,17,22,18,24,31,33,16,23,29,27,30,34,37,36);(6,8)(7,12,13,23)(9,17)(10,18)(11,16)(14,27,25,22)(15,24,26,34)(19,29)(20,31)(21,30)(28,36,35,33)(32,37)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 461);