Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C4×C5×#41 ⋊ C2 |
| Sylow | C5 ⋊ #3444 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Frattini | C22 | C24×D5 |
| Fitting | C4×C5×#41 | C2 |
| hypercenter | C4×#41 | D5 |
| hypocenter | C5 | #3444 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C42 ◁ C4×#41 | |
1212 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|d^2=g,e^2=f,b^a=bf,c^a=cg,h^b=h^4136 conjugacy classes — view the full character table.
| maximal subgroup of | #12014885, #12065809, #12065832, … |
| maximal normal subgroup of | #12014885, #12065809, #12065832, … |
| maximal quotient of | #12257599, #12257702, #12257706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19819);