Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C5×#520 ⋊ C2 |
| Sylow | C5 ⋊ C4×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator | C2×C5 | C24×C4 |
| Fitting | C5×#520 | C2 |
| socle, cosocle | C22×C5 | C25 |
| prime radical | C2×#4149 | C2 |
| hypercenter | #520 | D5 |
| hypocenter | C5 | C4×H(5,𝔽2) |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #520 | |
1660 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=g,e^2=g,b^a=bf,d^a=df,c^b=cf,h^b=h^4,h^c=h^4,h^e=h^4136 conjugacy classes — view the full character table.
| maximal subgroup of | #12105171, #12105177, #12105182, … |
| maximal normal subgroup of | #12105171, #12105177, #12105182, … |
| maximal quotient of | #12257532, #12257534, #12257538, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19869);