Groups of order 504

Factorization and useful forms
The order 504 has prime factors 2, 3, and 7, with factorization:

$$504 = 2^3 \cdot 3^2 \cdot 7 = 8 \cdot 9 \cdot 7$$

There are a number of useful forms for this number:

$$504 = 8^3 - 8 = 84(7 - 1)$$

Classification of non-solvable groups
First, we need to classify the possibilities for simple non-abelian composition factors. The only simple non-abelian groups of order dividing 504 are projective special linear group:PSL(3,2) (order 168, also isomorphic to $$PSL(2,7)$$) and projective special linear group:PSL(2,8) (order 504).

By order considerations, the possibilities for the composition factors are:


 * PSL(3,2) and cyclic group:Z3
 * PSL(2,8) alone

It remains to classify the groups with $$PSL(3,2)$$ and $$\mathbb{Z}_3$$ as its composition factors. It turns out that the only possibility is direct product of PSL(3,2) and Z3. We consider two (overlapping) possibilities:


 * There is a normal subgroup isomorphic to $$PSL(3,2)$$ and the quotient is cyclic of order three: In this, case, there is a mapping from the quotient group $$\mathbb{Z}_3$$ to the outer automorphism group of $$PSL(3,2)$$ via the conjugation action (see quotient group maps to outer automorphism group of normal subgroup). However, the outer automorphism group of $$PSL(3,2)$$ is isomorphic to cyclic group:Z2, which has no nontrivial homomorphisms from cyclic group:Z3. Thus, the outer action is trivial, and the whole group must therefore be an internal direct product, specifically, it must be direct product of PSL(3,2) and Z3.
 * There is a normal subgroup isomorphic to $$\mathbb{Z}_3$$ and the quotient group is isomorphic to $$PSL(3,2)$$: Note first that the normal subgroup must be in the center because the induced map $$PSL(3,2) \to \operatorname{Out}(\mathbb{Z}_3)$$ must be trivial. The possibilities for this can be judged by looking at $$H^2(PSL(3,2);\mathbb{Z}_3)$$ for the trivial action. But, in fact, $$PSL(3,2)$$ has Schur multiplier $$\mathbb{Z}_2$$ and trivial abelianization, so by the dual universal coefficients for group cohomology, the cohomology group math>H^2(PSL(3,2);\mathbb{Z}_3) is trivial. Thus, the only possible extension is direct product of PSL(3,2) and Z3.

GAP implementation
gap> SmallGroupsInformation(504);

There are 202 groups of order 504. They are sorted by their Frattini factors. 1 has Frattini factor [ 42, 1 ]. 2 has Frattini factor [ 42, 2 ]. 3 has Frattini factor [ 42, 3 ]. 4 has Frattini factor [ 42, 4 ]. 5 has Frattini factor [ 42, 5 ]. 6 has Frattini factor [ 42, 6 ]. 7 - 11 have Frattini factor [ 84, 7 ]. 12 - 18 have Frattini factor [ 84, 8 ]. 19 - 21 have Frattini factor [ 84, 9 ]. 22 has Frattini factor [ 84, 10 ]. 23 has Frattini factor [ 84, 11 ]. 24 - 28 have Frattini factor [ 84, 12 ]. 29 - 33 have Frattini factor [ 84, 13 ]. 34 - 38 have Frattini factor [ 84, 14 ]. 39 - 41 have Frattini factor [ 84, 15 ]. 42 has Frattini factor [ 126, 7 ]. 43 has Frattini factor [ 126, 8 ]. 44 has Frattini factor [ 126, 9 ]. 45 has Frattini factor [ 126, 10 ]. 46 has Frattini factor [ 126, 11 ]. 47 has Frattini factor [ 126, 12 ]. 48 has Frattini factor [ 126, 13 ]. 49 has Frattini factor [ 126, 14 ]. 50 has Frattini factor [ 126, 15 ]. 51 has Frattini factor [ 126, 16 ]. 52 has Frattini factor [ 168, 43 ]. 53 has Frattini factor [ 168, 44 ]. 54 has Frattini factor [ 168, 45 ]. 55 has Frattini factor [ 168, 46 ]. 56 has Frattini factor [ 168, 47 ]. 57 has Frattini factor [ 168, 48 ]. 58 has Frattini factor [ 168, 49 ]. 59 has Frattini factor [ 168, 50 ]. 60 has Frattini factor [ 168, 51 ]. 61 has Frattini factor [ 168, 52 ]. 62 has Frattini factor [ 168, 53 ]. 63 has Frattini factor [ 168, 54 ]. 64 has Frattini factor [ 168, 55 ]. 65 has Frattini factor [ 168, 56 ]. 66 has Frattini factor [ 168, 57 ]. 67 - 73 have Frattini factor [ 252, 26 ]. 74 has Frattini factor [ 252, 27 ]. 75 - 79 have Frattini factor [ 252, 28 ]. 80 - 84 have Frattini factor [ 252, 29 ]. 85 - 89 have Frattini factor [ 252, 30 ]. 90 has Frattini factor [ 252, 31 ]. 91 has Frattini factor [ 252, 32 ]. 92 - 98 have Frattini factor [ 252, 33 ]. 99 - 105 have Frattini factor [ 252, 34 ]. 106 - 110 have Frattini factor [ 252, 35 ]. 111 - 117 have Frattini factor [ 252, 36 ]. 118 - 122 have Frattini factor [ 252, 37 ]. 123 - 125 have Frattini factor [ 252, 38 ]. 126 has Frattini factor [ 252, 39 ]. 127 has Frattini factor [ 252, 40 ]. 128 - 132 have Frattini factor [ 252, 41 ]. 133 - 137 have Frattini factor [ 252, 42 ]. 138 - 142 have Frattini factor [ 252, 43 ]. 143 - 147 have Frattini factor [ 252, 44 ]. 148 - 152 have Frattini factor [ 252, 45 ]. 153 - 155 have Frattini factor [ 252, 46 ]. 156 - 202 have trivial Frattini subgroup.

For the selection functions the values of the following attributes are precomputed and stored: IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup, LGLength, FrattinifactorSize and FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library. IdSmallGroup is available for this size.