Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10619373 |
| commutator, prime radical, hypocenter | #564⋊C5 | C4 |
| Frattini, hypercenter | C22 | #66297 |
| Fitting | #564 | AGL(1,𝔽5) |
| cosocle | #10617013 | C2 |
| LCS | G ▷ #564⋊C5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^2,5,2^6|a^2=b,b^2=i,d^2=i,e^2=h,f^2=hi,g^2=i,c^a=c^2,d^a=de,e^a=fg,f^a=ehi,g^a=ghi,h^a=hi,c^b=c^4,d^b=defg,e^b=eg,f^b=fgh,g^b=gi,d^c=g,e^c=dg,f^c=degh,g^c=defgi,e^d=eh,f^d=fhi,g^d=gh,f^e=fhi,g^e=gi,g^f=gi20 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116362);