Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
332 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 37 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=d,d^2=ei,g^2=i,h^2=i,b^a=bc,c^a=ce,f^a=f^2,g^a=gh,h^a=hi,d^b=de,g^f=h,h^f=ghi,h^g=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72258519, #72268912 |
| maximal normal subgroup of | #72258519, #72268912 |
| maximal quotient of | #72228215, #72268921 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085410);