Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C5 ⋊ M128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C2×C32 |
| Fitting | C5×C32 | C4 |
| prime radical | C2×D5 | C32 |
| cosocle | #3926 | C22 |
| hypercenter | C32 | AGL(1,𝔽5) |
| hypocenter | C5 | M128 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C16 ◁ C32 | |
38 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,b^a=bg,h^a=h^2,h^b=h^4,h^c=h^4112 conjugacy classes — view the full character table.
| maximal subgroup of | #13014522, #13014523, #13014588, … |
| maximal normal subgroup of | #13014522, #13014523, #13014588, … |
| maximal quotient of | #13014521, #13014527, #13014593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 566);