Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C2×C5×#560 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1344 conjugacy classes of subgroups; only the 776 normal subgroups are shown.
This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 776 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 — 5 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4)(5,12)(6,11)(7,9)(8,14)(10,16)(13,17)(15,18);(3,5)(4,8)(6,14)(7,15)(9,17)(10,18)(11,12)(13,16);(4,9)(5,13)(8,12)(10,11)(14,15)(17,18);(4,10)(5,14)(8,17)(9,11)(12,18)(13,15)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21392);