Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | #564 ⋊ D5 |
| derived | #564⋊C5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | (C24⋊C5)⋊C2 |
| commutator, cosocle, hypocenter | #564⋊C5 | C2 |
| Fitting | #564 | D5 |
| LCS | G ▷ #564⋊C5 | |
| UCS | C1 ◁ C22 | |
77 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2,5,2^6|c^2=g,d^2=g,e^2=gh,f^2=h,b^a=b^4,c^a=dg,d^a=cg,e^a=def,f^a=cdfg,c^b=d,d^b=de,e^b=ef,f^b=cfg,d^c=dgh,e^c=eg,f^c=fg,e^d=egh,f^d=fh,f^e=fh22 conjugacy classes — view the full character table.
| maximal subgroup of | #13046093, C2×#10617014, #13046122, … |
| maximal normal subgroup of | #13046093, C2×#10617014, #13046122, … |
| maximal quotient of | #13046097, C2×#10617014, C3×#10617014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19096);