Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C2×C5×#458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C2×C5×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C5 ◁ G | |
794 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 794 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 — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,7,10)(5,8,12,15)(6,9,13,16)(11,17,18,14);(3,5)(4,8)(6,11)(7,12)(9,14)(10,15)(13,18)(16,17);(3,6)(4,9)(5,11)(7,13)(8,14)(10,16)(12,18)(15,17)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18701);