Finite Group DatabasebetaFinite Group Databasebeta| order | 252 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C32×D3 (order 216) · Inn(G) = C1 · Out(G) = C22×C32×D3 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C22×C7×C9 |
| multiplicative | (ℤ/381)ˣ, (ℤ/508)ˣ, (ℤ/762)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C22×C3×C7 |
| socle, prime radical | C22×C3×C7 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20);(3,4)252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(252, 15);