Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#1827 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #117 |
| commutator | C2×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | #381 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C5×#117 ◁ G | |
262 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 21 points:
(1,2,3,4,5)(6,7)(8,15,17,11)(9,14)(10,13)(12,16,19,21)(18,20);(6,8)(7,11)(9,16)(10,17)(12,14)(13,15)(18,21)(19,20);(6,8)(7,12,13,19)(9,16)(10,17)(11,14,15,20)(18,21)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5417);