Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C5×C32 ⋊ C4 |
| Sylow | C5 ⋊ #1397 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C4×C16 |
| Fitting | C5×C32 | C4 |
| prime radical | C2×D5 | C2×C16 |
| cosocle | D5×C16 | C22 |
| hypercenter | C32 | AGL(1,𝔽5) |
| hypocenter | C5 | #1397 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C16 ◁ C32 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=c,b^2=d,c^2=e,e^2=f,f^2=g,b^a=bg,h^b=h^2,h^d=h^4112 conjugacy classes — view the full character table.
| maximal subgroup of | #13014522, #13014523, C2×#10598486, … |
| maximal normal subgroup of | #13014522, #13014523, C2×#10598486, … |
| maximal quotient of | #13014064, #170⋊C32, (C2×AGL(1,𝔽5))⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 568);