Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #226⋊C16 |
129 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 31 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:
2^5,3,2^3|a^2=i,b^2=c,c^2=d,d^2=e,g^2=i,h^2=i,b^a=bi,f^a=f^2,g^a=gh,h^a=hi,g^f=h,h^f=ghi,h^g=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085765);