Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C4×C5×Q8 ⋊ C4 |
| Sylow | C5 ⋊ #2355 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting | C4×C5×Q8 | C4 |
| prime radical, cosocle | C22×D5 | C24 |
| hypercenter | C4×Q8 | AGL(1,𝔽5) |
| hypocenter | C5 | #2355 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
480 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=g,b^2=ef,c^2=f,d^2=e,b^a=be,c^a=cf,h^a=h^2,d^b=de,h^g=h^470 conjugacy classes — view the full character table.
| maximal subgroup of | #13018973, #13018977, #13018981, … |
| maximal normal subgroup of | #13018973, #13018977, #13018981, … |
| maximal quotient of | C4⋊C4×C4⋊AGL(1,𝔽5), #13028007, #13028021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19295);