Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#2955 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C2×D4 |
| commutator | C4 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| socle | C2×C5 | C2×C4×D4 |
| prime radical | C5×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
456 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)(9,14,22,30,23,29,21,34)(18,31,32,26,33,37,35,36);(6,8)(7,13)(9,18)(10,25)(11,19)(12,20)(14,26)(15,27)(16,28)(17,24)(21,35)(22,32)(23,33)(29,36)(30,37)(31,34);(6,9,11,22,12,23,10,21)(7,14,16,30,15,29,24,34)(8,18,19,32,20,33,25,35)(13,26,28,37,27,36,17,31);(7,15)(13,27)(14,29)(16,24)(17,28)(26,36)(30,34)(31,37)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18632);