Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3036 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,6,7,8)(10,15)(11,19)(12,13)(16,20)(17,24)(21,23);(2,5)(3,6)(7,8)(9,10,12,22,14,15,13,18)(11,20,19,24,21,17,23,16);(1,3,4,7)(2,6,5,8)(9,11)(10,16,15,24)(12,23)(13,19)(14,21)(17,18,20,22);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,23)(13,19)(14,21)(15,20)(16,18)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=efi,c^2=e,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253374);