Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C5×H(3,𝔽4) ⋊ C2 |
| Sylow | C5 ⋊ #3568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D5 |
| hypocenter | C5 | #3568 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
958 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 958 conjugacy classes of subgroups — too many to draw. Only its 449 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=f,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^c=h^4,h^e=h^476 conjugacy classes — view the full character table.
| maximal subgroup of | #12080947, #12080955, #12080962, … |
| maximal normal subgroup of | #12080947, #12080955, #12080962, … |
| maximal quotient of | #12262634, #12263591, #12263676, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21071);