Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C2×(C24⋊C5)⋊C2 |
| commutator, cosocle, hypocenter | #564⋊C5 | C22 |
| Fitting | #3531 | D5 |
| prime radical | #3531⋊C5 | C2 |
| hypercenter | C2×C4 | (C24⋊C5)⋊C2 |
| LCS | G ▷ #564⋊C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
252 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
2,5,2^7|a^2=i,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=h,b^a=b^4,c^a=cdef,d^a=df,e^a=efh,f^a=fi,g^a=gh,c^b=ce,d^b=dfh,e^b=cef,f^b=cdi,d^c=dh,e^c=ehi,f^c=fh,e^d=ehi,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116396);