Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×#1336 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #102 |
| commutator | C8 | C2×C5×C8 |
| Frattini, cosocle | C4×C8 | C22×C5 |
| socle | C2×C5 | D4⋊C8 |
| prime radical | C5×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ C4×C5×C8 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7,10,14,11,15,21,20,12,16,9,18,19,8,17,13);(6,8)(7,11)(9,13)(10,20)(12,15)(14,17)(16,19)(18,21)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 469);