Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#3025 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| socle | C2×C5 | C2×#120 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
766 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 212 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)(11,17)(18,21);(6,7)(8,11)(9,12)(10,13)(14,18)(15,17)(16,19)(20,21);(6,8)(7,11)(9,14)(10,15)(12,18)(13,17)(16,20)(19,21);(6,9,10,16)(7,12,13,19)(8,14,15,20)(11,18,17,21)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18702);