Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | C9⋊#564 |
| Sylow | C9 ⋊ #564 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D9 |
| commutator, cosocle | C22×C9 | C24 |
| Frattini, socle, prime radical | C22×C3 | C23×D3 |
| Fitting | C9×#125 | C2 |
| hypercenter | #125 | D9 |
| hypocenter | C9 | #564 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ #125 | |
258 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^2|a^2=f,b^2=e,c^2=e,d^2=f,g^3=h,c^a=cf,d^a=de,g^a=g^2h^2,h^a=h^2,c^b=cef,d^b=df75 conjugacy classes — view the full character table.
| maximal subgroup of | #11780540, #11780546, #11780699, … |
| maximal normal subgroup of | #11780540, #11780546, #11780699, … |
| maximal quotient of | #11807974, #11807977, #11807980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1670);