Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | Aut(G) = #11892456 (order 1152) · Inn(G) = AGL(1,𝔽9) · Out(G) = #41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C2×C16 |
| Frattini | C2 | C2×AGL(1,𝔽9) |
| Fitting | C32×C4 | C8 |
| socle | C2×C32 | C2×C8 |
| prime radical | C2×C3⋊D3 | C8 |
| cosocle | #622 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,4,9)(2,7,3)(5,8,6)(10,12,15,14)(11,16,19,18)(13,20,22,21)(17,23,25,24);(2,3,4,7,5,8,9,6)(10,11,13,17,14,18,21,24,15,19,22,25,12,16,20,23)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=e,c^2=d,d^2=e,f^a=g,g^a=fg^2,f^c=fg^2,g^c=f^2g^2,f^d=f^2,g^d=g^236 conjugacy classes — view the full character table.
| maximal subgroup of | #10593160, #10593161, #10593254, … |
| maximal normal subgroup of | #10593160, #10593161, #10593254, … |
| maximal quotient of | #10593248, #10593249, C4×#3771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 861);