Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | #386 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,12,14,13,16,11,15)(17,18,20,23,21,19,22,24);(10,14)(11,12)(15,16)(17,19,20,24,21,18,22,23);(1,2,4,6)(3,8,7,5)(10,14)(11,12)(15,16)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=h,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6793);