Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#2436 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C2×C5×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
852 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 248 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) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(11,13)(14,15,18,21)(16,20,17,19);(7,9)(11,13)(14,16,18,17)(15,19,21,20);(6,7)(8,9)(10,11,12,13)(14,15)(16,19)(17,20)(18,21);(14,17,18,16)(15,20,21,19)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18113);