Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
665 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 665 conjugacy classes of subgroups — too many to draw. Only its 111 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:
2^5,3,2^3|a^2=e,c^2=d,e^2=i,g^2=i,h^2=i,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,g^f=h,h^f=ghi,h^g=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226116, #72226188, #72258634, … |
| maximal normal subgroup of | #72226116, #72226188, #72258634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086935);