Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(9,10,14,19,12,18,21,15,13,17,23,24,22,16,11,20);(1,2,4,3)(5,6)(7,8)(9,11,12,14,13,21,22,23)(10,15,18,24,17,20,16,19);(1,3,4,2)(6,8)(10,16,17,18)(11,21)(14,23)(15,24,20,19)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56120);