Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #264427 |
| 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 | #10228 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
751 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)(10,14)(11,16)(12,15)(18,21)(22,24);(1,3,4,8)(2,5,6,7)(9,10,15,16,13,11,12,14)(18,22)(20,23)(21,24);(2,6)(5,7)(10,11)(14,16)(17,18)(19,21)(20,24)(22,23);(2,6)(5,7)(9,11,12,16,13,10,15,14)Polycyclic presentation:
2^9|b^2=eh,d^2=efi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 220399);