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, socle, hypercenter | C22 | #66297 |
| commutator, hypocenter | 2−1+4⋊C5 | C8 |
| Fitting | C2×2−1+4 | AGL(1,𝔽5) |
| prime radical | C2×2−1+4⋊C5 | C4 |
| cosocle | #10617016 | C2 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C22 | |
108 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^2,5,2^6|a^2=b,b^2=i,e^2=h,f^2=h,c^a=c^2,d^a=de,e^a=fg,f^a=eh,g^a=gh,c^b=c^4,d^b=defg,e^b=eg,f^b=fgh,d^c=g,e^c=dg,f^c=degh,g^c=defg,e^d=eh,f^d=fh,g^d=gh,f^e=fh32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116359);