Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #346 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×AGL(1,𝔽9) |
| commutator, socle | C2×C32 | C4×C8 |
| Fitting | C32×C8 | C8 |
| prime radical | C2×C3⋊D3 | C2×C8 |
| cosocle | #3787 | C22 |
| hypercenter | C8 | AGL(1,𝔽9) |
| hypocenter | C32 | #346 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C8 | |
65 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^2|a^2=c,b^2=d,c^2=e,d^2=f,e^2=f,b^a=bf,g^a=g^2h^2,h^a=g^2,g^c=g^2h,h^c=gh,g^e=g^2,h^e=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #11771333, #11771562, #11771613, … |
| maximal normal subgroup of | #11771333, #11771562, #11771613, … |
| maximal quotient of | #11771304, #11771316, #11772269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5137);