Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#339 ⋊ C2 |
| Sylow | C5 ⋊ #2002 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #4028 |
| commutator, cosocle | C22×C4×C5 | C23 |
| Frattini | C22×C4 | C22×D5 |
| Fitting | C5×#339 | C2 |
| socle | C22×C5 | #120 |
| hypercenter | #339 | D5 |
| hypocenter | C5 | #2002 |
| LCS | G ▷ C22×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #339 | |
478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^7,5|d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fg,h^a=h^4,c^b=cf,d^b=dg,h^b=h^4,d^c=dg,h^c=h^482 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#2002, C2×#10599554, #12126210, … |
| maximal normal subgroup of | D5×#2002, C2×#10599554, #12126210, … |
| maximal quotient of | #11999499, #11999541, #11999555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 1636);