Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of GL(3,𝔽2) |
| automorphism | Aut(G) = C2×SO(3,𝔽7) (order 672) · Inn(G) = GL(3,𝔽2) · Out(G) = C22 |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
| direct product | C3×GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C3 | GL(3,𝔽2) |
| commutator, hypocenter | GL(3,𝔽2) | C3 |
| Frattini | C1 | G |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C3 | |
35 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 21 |
A minimal faithful permutation representation — 2 generators on 10 points:
(4,5)(7,8);(1,2,3)(5,6,7)(8,9,10)18 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 157);