Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#1280 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22⋊C4 |
| commutator | C22 | C4×C5×C8 |
| Frattini, cosocle | C23×C4 | C22×C5 |
| prime radical | C24×C5 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
264 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5)(10,11)(12,16,17,14)(13,15)(18,19,20,22,21,23,24,25);(6,7,8,9)(10,12)(11,14)(13,17)(15,16)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 413);