Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C35 |
| unambiguous semidirect product | C3⋊(C3⋊#272) |
| Fitting, derived, Sylow | C32×H(3,𝔽3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C3⋊(C3⋊(C3⋊D3)) |
| commutator, Fitting, cosocle, hypocenter | C32×H(3,𝔽3) | C2 |
| socle | C33 | C3⋊D3 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 5 generators on 15 points:
(2,3)(5,6)(8,10)(11,15)(12,14);(8,11,12)(10,14,15);(7,8,10)(9,12,14)(11,15,13);(4,5,6);(1,2,3)Polycyclic presentation:
2,3^5|b^a=b^2,c^a=c^2,d^a=d^2,e^a=e^2,e^d=ef54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 248);