Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
305 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^5,3,2^3|a^2=i,b^2=c,d^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=f^2,g^a=gh,h^a=hi,d^b=dei,g^f=h,h^f=ghi,h^g=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085603);