Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | #66297 |
| commutator, prime radical, hypocenter | 2−1+4⋊C5 | C4 |
| Fitting | 2−1+4 | AGL(1,𝔽5) |
| cosocle | #66243 | C2 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C2 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^2,5,2^5|a^2=b,b^2=h,d^2=h,e^2=h,f^2=h,c^a=c^2,d^a=efg,e^a=fh,f^a=d,g^a=defh,c^b=c^4,d^b=eh,e^b=dh,f^b=efg,g^b=degh,d^c=e,e^c=ef,f^c=fg,g^c=dgh,e^d=eh,f^d=fh,g^d=gh,f^e=fh16 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10619372, #13046112, #13046113, … |
| maximal normal subgroup of | C2×#10619372, #13046112, #13046113, … |
| maximal quotient of | #13046091, C2×#10619372, C3×#10619372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21454);