Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #625 |
| commutator | C32×C4 | C2×C8 |
| Frattini, hypercenter | C42 | D32 |
| Fitting, cosocle | C32×C42 | C22 |
| socle, prime radical | C22×C32 | C22⋊C4 |
| hypocenter | C32 | Q8⋊C8 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
162 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^2|a^2=f,b^2=c,c^2=e,d^2=f,b^a=bd,d^a=df,h^a=h^2,d^b=df,g^b=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#4484, D3×#4535, C2×#10589083, … |
| maximal normal subgroup of | D3×#4484, D3×#4535, C2×#10589083, … |
| maximal quotient of | #11770677, #11770691, #11770794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 965);