Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | 2−1+4⋊C5 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | (C24⋊C5)⋊C2 |
| commutator, hypocenter | 2−1+4⋊C5 | C8 |
| Fitting | C4×2−1+4 | D5 |
| socle | C22 | #66245 |
| prime radical | C2×2−1+4⋊C5 | C4 |
| cosocle | C4×2−1+4⋊C5 | C2 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C2×C4 | |
198 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 198 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=h,d^2=g,e^2=g,h^2=i,b^a=b^4,c^a=cdef,d^a=df,e^a=efg,c^b=f,d^b=cf,e^b=cdfg,f^b=cdef,d^c=dg,e^c=eg,f^c=fg,e^d=eg56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116363);