Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #2627 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C2×C5×C22⋊Q8 | C2 |
| prime radical | C24×C5 | C23 |
| hypercenter | C2×C22⋊Q8 | D5 |
| hypocenter | C5 | #2627 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊Q8 | |
508 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=f,b^2=e,c^2=f,b^a=be,c^a=cf,c^b=cg,h^b=h^4,d^c=de88 conjugacy classes — view the full character table.
| maximal subgroup of | #11995345, #11995355, #11999556, … |
| maximal normal subgroup of | #11995345, #11995355, #11999556, … |
| maximal quotient of | #12025719, #12025734, #12025842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 16601);