Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C5×#125 ⋊ C2 |
| Sylow | C5 ⋊ #3553 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C2×C5×#125 | C2 |
| hypercenter | C2×#125 | D5 |
| hypocenter | C5 | #3553 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×#125 | |
932 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|b^2=g,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,h^a=h^4,c^b=cg,d^b=df94 conjugacy classes — view the full character table.
| maximal subgroup of | #11991849, #11991852, #12081167, … |
| maximal normal subgroup of | #11991849, #11991852, #12081167, … |
| maximal quotient of | #12262191, #12262420, #12262873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 20085);