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, hypercenter | C2×C4 | (C24⋊C5)⋊C2 |
| commutator, hypocenter | #564⋊C5 | C4 |
| Fitting | #3531 | D5 |
| socle | C22 | #66245 |
| prime radical, cosocle | #3531⋊C5 | C2 |
| LCS | G ▷ #564⋊C5 | |
| UCS | C1 ◁ C2×C4 | |
156 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 13 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=h,c^2=i,d^2=g,e^2=gi,f^2=i,h^2=i,b^a=b^4,c^a=cdef,d^a=df,e^a=efg,f^a=fi,c^b=f,d^b=cf,e^b=cdfg,f^b=cdefi,d^c=dg,e^c=egi,f^c=fg,e^d=egi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116364);