Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#2883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C23 | C24×C5 |
| prime radical | C2×C5×H(5,𝔽2) | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5×D4 ◁ G | |
800 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 340 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 — 4 generators on 21 points:
(1,2,3,4,5)(6,7)(8,14,16,11)(9,12)(10,13)(15,20,21,18)(17,19);(6,8)(7,11)(9,15)(10,16)(12,18)(13,14)(17,21)(19,20);(6,9)(7,12)(8,15)(10,17)(11,18)(13,19)(14,20)(16,21);(9,17)(12,19)(15,21)(18,20)205 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18560);