Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#338 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C42 |
| commutator | C4 | C2×C42×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C22×C5 | #95 |
| prime radical | C24×C5 | Q8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
372 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,6,10,8,11,7,16)(5,12,14,9,15,17,13,18);(3,5,7,14,8,15,6,13)(4,9,10,12,11,18,16,17)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5321);