Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C4×C5×D8 ⋊ C2 |
| Sylow | C5 ⋊ #3325 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C4×C5×D8 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | C4×D8 | D5 |
| hypocenter | C5 | #3325 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×D8 | |
460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=g,f^2=g,b^a=beg,c^a=cf,d^a=dg,f^a=fg,h^a=h^4,c^b=cg,d^b=deg,h^b=h^4,f^c=fg,h^d=h^485 conjugacy classes — view the full character table.
| maximal subgroup of | #12088887, #12088889, #12088947, … |
| maximal normal subgroup of | #12088887, #12088889, #12088947, … |
| maximal quotient of | #12114917, #12115762, #12116621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 13425);