Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #13400 |
402 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 93 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|c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^b=eh,f^b=fh,d^c=dh,e^c=egh,f^c=fh,g^c=gh,i^c=i^266 conjugacy classes — view the full character table.
| maximal subgroup of | #72270131, #72270166, #72270170, … |
| maximal normal subgroup of | #72270131, #72270166, #72270170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88860);