Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽7) (order 504) · Inn(G) = AGL(1,𝔽7) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C7⋊C8)⋊C9 |
| unambiguous semidirect product | (C7⋊C9)⋊C8 |
| derived, Sylow | C7 ⋊ C8×C9 |
| Hall | C7⋊C9 ⋊ C8 |
| Hall | C7⋊C8 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C4 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C8×C9 |
| Fitting, cosocle | C3×C4×C7 | C2×C3 |
| socle, prime radical | C2×C3×C7 | C3×C4 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C4 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,6,5,3,4,7)(8,9,10,12,11,13,14,15)(16,17,19,18,20,22,21,23,24);(1,2,5,6,4,7,3)Polycyclic presentation:
2,3,2,3,2,7|a^2=c,b^3=d,c^2=e,f^a=f^6,f^b=f^484 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D7⋊C9, #1131⋊C9, C2×(C7⋊C8)⋊C9, … |
| maximal normal subgroup of | C8×D7⋊C9, #1131⋊C9, C2×(C7⋊C8)⋊C9, … |
| maximal quotient of | (C7⋊C9)⋊C16, C2×(C7⋊C8)⋊C9, (C7⋊C8)⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 1);