Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #29387 |
646 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 646 conjugacy classes of subgroups — too many to draw. Only its 141 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|b^2=e,d^2=g,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh,i^d=i^266 conjugacy classes — view the full character table.
| maximal subgroup of | #72270136 |
| maximal normal subgroup of | #72270136 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 272049);