Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #12266 |
592 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 59 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=f,c^2=fg,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,i^a=i^2,c^b=ch,d^b=df,e^b=egh,d^c=dg,e^c=eh,e^d=eh60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 82686);