Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #332071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,4,7,6)(9,10,11,13)(12,14,16,15)(17,18,20,21)(19,23,24,22);(1,3,5,7)(2,6,8,4)(10,13)(14,15)(17,19)(18,22)(20,24)(21,23);(1,4,5,6)(2,7,8,3)(10,14)(12,16)(13,15)(18,21)(19,23)(22,24);(2,8)(3,7)(10,15)(12,16)(13,14)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=ehi,c^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 246191);