Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | GO+(4,𝔽2)⋊Q8 |
| Sylow | C32 ⋊ #476 |
135 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^2|a^2=d,c^2=e,d^2=f,b^a=bdef,c^a=ce,g^a=g^2h^2,h^a=g^2,d^b=df,g^b=gh^2,h^b=h^2,g^d=g^2h,h^d=gh,g^f=g^2,h^f=h^227 conjugacy classes — view the full character table.
| maximal subgroup of | GO+(4,𝔽2)⋊SD16, #11779400, #11779557, … |
| maximal normal subgroup of | GO+(4,𝔽2)⋊SD16, #11779400, #11779557, … |
| maximal quotient of | #11785194, #11786955, #11789168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8285);