Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | Aut(G) = AGL(1,𝔽9)⋊C8 (order 576) · Inn(G) = AGL(1,𝔽9) · Out(G) = C8 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C32 |
| Fitting | C32×C4 | C8 |
| socle, prime radical | C2×C32 | C16 |
| cosocle | #3708 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 41 points:
(2,3,4,7,5,8,9,6)(10,11,12,16,13,17,20,26,14,18,21,27,23,29,32,36,15,19,22,28,24,30,33,37,25,31,34,38,35,39,40,41);(1,4,9)(2,7,3)(5,8,6)Polycyclic presentation:
2^5,3^2|a^2=b,b^2=c,c^2=d,d^2=e,f^a=g,g^a=fg^2,f^b=fg^2,g^b=f^2g^2,f^c=f^2,g^c=g^236 conjugacy classes — view the full character table.
| maximal subgroup of | #10593248, #10593249, C2×#63725, … |
| maximal normal subgroup of | #10593248, #10593249, C2×#63725, … |
| maximal quotient of | #10590011, C2×#63725, #11720689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 373);