Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #7491 |
276 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|a^2=d,b^2=e,c^2=h,d^2=h,f^2=h,b^a=bc,c^a=cf,f^a=fg,g^a=gh,i^a=i^2,c^b=ch,d^b=df,f^b=fh,g^b=gh,i^b=i^2,d^c=dgh,f^c=fh,f^d=fh72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 57326);