Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #334823 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 | |
692 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)(3,7)(4,6)(5,8)(9,10,12,15,13,16,14,11);(1,3)(2,5)(4,8)(6,7)(18,21)(19,23)(22,24);(9,11)(10,14)(12,16)(13,15)(17,18,20,22)(19,24,23,21);(2,6)(3,8)(9,10,13,16)(11,14,15,12)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 249440);