Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(9,10,14,16,12,11,13,15)(17,18,20,23)(19,21,24,22);(5,6,7,8)(13,14)(15,16)(18,21)(19,24)(22,23);(1,2,3,4)(9,11,12,10)(13,15,14,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=i,e^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=ehi,f^c=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17366);