Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#1763 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D4 |
| commutator | C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
324 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,13,12)(14,15)(16,17)(18,20)(19,21);(7,9)(14,16,18,21,19,20,17,15);(10,12,13,11)(14,17,19,18)(15,16,21,20)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5353);