Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×#1321 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #99 |
| commutator | C2×C4 | C2×C5×C8 |
| Frattini, cosocle | C2×M16 | C22×C5 |
| socle | C2×C5 | #323 |
| prime radical | C2×C5×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×M16 ◁ G | |
112 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)(6,7,10,15,11,14,25,31,12,16,26,33,9,21,23,32)(8,17,19,34,18,13,27,28,20,30,24,36,22,29,35,37);(6,8)(7,13)(9,22)(10,24)(11,18)(12,20)(14,30)(15,28)(16,29)(17,21)(19,26)(23,27)(25,35)(31,36)(32,34)(33,37)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 454);