Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | #564⋊C5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C23 | (C24⋊C5)⋊C2 |
| commutator, hypocenter | #564⋊C5 | C4 |
| Fitting | C2×#564 | D5 |
| prime radical, cosocle | C2×#564⋊C5 | C2 |
| LCS | G ▷ #564⋊C5 | |
| UCS | C1 ◁ C23 | |
186 conjugacy classes of subgroups.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
2,5,2^7|a^2=i,c^2=h,d^2=g,e^2=gh,f^2=h,b^a=b^4,c^a=cdef,d^a=df,e^a=efg,f^a=fh,c^b=f,d^b=cf,e^b=cdfg,f^b=cdefh,d^c=dg,e^c=egh,f^c=fg,e^d=egh,f^d=fh,f^e=fh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72209975 |
| maximal normal subgroup of | #72209975 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116365);