Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C5×#557 ⋊ C2 |
| Sylow | C5 ⋊ Q8×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×#557 | C2 |
| prime radical | C2×#713 | C22 |
| hypercenter | #557 | D5 |
| hypocenter | C5 | Q8×#41 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #557 | |
1068 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^b=df,h^b=h^4,h^d=h^4,h^e=h^4100 conjugacy classes — view the full character table.
| maximal subgroup of | #12075279, #12075287, #12075294, … |
| maximal normal subgroup of | #12075279, #12075287, #12075294, … |
| maximal quotient of | C4⋊C4×#713, #12258617, #12258727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 20998);