Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #2258 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C22×C16 |
| Fitting | C5×#131 | C4 |
| prime radical | C5×#41 | C8 |
| cosocle | #672 | C23 |
| hypercenter | #131 | AGL(1,𝔽5) |
| hypocenter | C5 | #2258 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C8 ◁ #131 | |
106 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=d,d^2=e,e^2=f,f^2=g,h^a=h^2,c^b=cg,h^d=h^4100 conjugacy classes — view the full character table.
| maximal subgroup of | #13027845, #13027846, #13027848, … |
| maximal normal subgroup of | #13027845, #13027846, #13027848, … |
| maximal quotient of | #13027527, #13027548, #13027793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 6856);