Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #458⋊C9 |
| Sylow | #458 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C3 | #120⋊C3 |
| commutator, hypocenter | H(5,𝔽2) | C2×C9 |
| Frattini | C23×C3 | C2×A4 |
| Fitting | C3×#458 | C3 |
| prime radical, cosocle | C3×H(5,𝔽2) | C2×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C3 | |
79 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^2,2^5|a^2=h,b^3=c,d^a=df,e^a=eg,d^b=e,e^b=de,f^b=g,g^b=fgh,g^d=gh,f^e=fh48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#458⋊C9, #3025⋊C9, #3026⋊C9, … |
| maximal normal subgroup of | C2×#458⋊C9, #3025⋊C9, #3026⋊C9, … |
| maximal quotient of | #1790⋊C9, #2030⋊C9, #2062⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 205);