Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#1390 ⋊ C2 |
| Sylow | C3 ⋊ #13047 |
888 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 89 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,b^2=df,e^2=h,b^a=bd,c^a=ce,e^a=egh,g^a=gh,e^b=egh,i^b=i^2,d^c=dg,e^c=eh,f^c=fg,i^c=i^2,e^d=eh,f^e=fh66 conjugacy classes — view the full character table.
| maximal subgroup of | #72248393, #72248456, #72248498, … |
| maximal normal subgroup of | #72248393, #72248456, #72248498, … |
| maximal quotient of | #72255077 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 86869);