Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C5 ⋊ #2642 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, prime radical, cosocle | C23×C5 | C24 |
| Fitting | C2×C5×#125 | C2 |
| hypercenter | C2×#125 | D5 |
| hypocenter | C5 | #2642 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=f,c^2=eg,d^2=eg,b^a=beg,c^a=cf,h^a=h^4,c^b=cg,h^b=h^4,d^c=deg,h^c=h^488 conjugacy classes — view the full character table.
| maximal subgroup of | #12000689, #12000693, #12000745, … |
| maximal normal subgroup of | #12000689, #12000693, #12000745, … |
| maximal quotient of | #12025709, #12025835, #12025906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 16856);