Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#1316 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #99 |
| commutator | C23 | C2×C5×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C22×C5 |
| prime radical | C25×C5 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C2×C5×C22⋊C4 ◁ G | |
294 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7,8,12,9,13,15,19)(10,11,14,17,16,18,20,21);(7,11)(13,18)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 449);