Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | 2−1+4⋊C5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | (C24⋊C5)⋊C2 |
| commutator, hypocenter | 2−1+4⋊C5 | C4 |
| Fitting | #585 | D5 |
| socle | C2 | #66245 |
| prime radical, cosocle | #585⋊C5 | C2 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C4 | |
121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
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^4,d^a=eh,e^a=dh,f^a=efg,g^a=degh,d^c=e,e^c=ef,f^c=fg,g^c=dgh,e^d=eh,f^d=fh,g^d=gh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #13046092, C2×#10617017, #13046138, … |
| maximal normal subgroup of | #13046092, C2×#10617017, #13046138, … |
| maximal quotient of | #13046095, #13046096, C2×#10617017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19099);