Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C5×#543 ⋊ C2 |
| Sylow | C5 ⋊ #3561 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×#543 | C2 |
| hypercenter | #543 | D5 |
| hypocenter | C5 | #3561 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #543 | |
1032 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,h^b=h^4,h^d=h^4,h^e=h^482 conjugacy classes — view the full character table.
| maximal subgroup of | #12072653, #12072659, #12072664, … |
| maximal normal subgroup of | #12072653, #12072659, #12072664, … |
| maximal quotient of | #12262237, #12262306, #12262549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 20559);