Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #12766 |
760 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 127 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,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^2,c^b=ch,d^b=dh,e^b=egh,f^b=fgh,d^c=dg,e^d=eh,f^d=fh72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#12766, #72247871, #72247954, … |
| maximal normal subgroup of | D3×#12766, #72247871, #72247954, … |
| maximal quotient of | #72254130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 85408);