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, socle, hypercenter | C22 | (C24⋊C5)⋊C2 |
| commutator, hypocenter | 2−1+4⋊C5 | C4 |
| Fitting | C2×2−1+4 | D5 |
| prime radical, cosocle | C2×2−1+4⋊C5 | C2 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C22 | |
131 conjugacy classes of subgroups.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^2,5,2^5|a^2=b,e^2=h,f^2=h,c^a=c^4,d^a=g,e^a=eh,f^a=defgh,g^a=d,d^c=df,e^c=de,f^c=efgh,g^c=d,f^d=fh,g^d=gh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #13046091, C2×#10617016, #13046134, … |
| maximal normal subgroup of | #13046091, C2×#10617016, #13046134, … |
| maximal quotient of | #13046095, #13046096, #13046097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19098);