Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C5×#489 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
436 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 320 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18);(4,7)(8,10)(11,13,16,18,17,14,15,12);(3,5,6,9)(4,8,7,10)(11,14,15,18,17,13,16,12)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18809);