Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (Q8⋊C9)⋊C8 |
| derived | Q8⋊C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | Q8⋊C9 | C8 |
| Frattini | C2×C3×C4 | S4 |
| Fitting | C3×C4×Q8 | D3 |
| socle, prime radical | C22×C3 | SL(2,ℤ/4) |
| cosocle | C4×Q8⋊C9 | C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2×C4 | |
69 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^3,3^2,2^3|a^2=b,b^2=c,d^3=e^2,f^2=h,g^2=h,d^a=d^2e,e^a=e^2,f^a=g,g^a=f,f^d=gh,g^d=fg,g^f=gh60 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#3688, #3688⋊C8, #11919964, … |
| maximal normal subgroup of | C8×#3688, #3688⋊C8, #11919964, … |
| maximal quotient of | (Q8⋊C9)⋊C16, C2×(Q8⋊C9)⋊C8, (Q8⋊C27)⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 186);