Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C5×Q82 ⋊ C2 |
| Sylow | C5 ⋊ #3522 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D5 |
| commutator, socle, cosocle | C22×C5 | C25 |
| Fitting | C5×Q82 | C2 |
| hypercenter | Q82 | D5 |
| hypocenter | C5 | #3522 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
1104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|b^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,h^a=h^4,d^b=df,h^c=h^4,h^e=h^494 conjugacy classes — view the full character table.
| maximal subgroup of | #12075769, #12075780, #12075857, … |
| maximal normal subgroup of | #12075769, #12075780, #12075857, … |
| maximal quotient of | #12259536, #12260061, #12260165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21026);