Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#127 |
| prime radical | #10304 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10,11,15,13,16,12,14);(2,5)(6,8)(10,14)(11,12)(15,16)(17,18)(19,21)(20,23)(22,24);(10,15)(11,12)(14,16)(18,21)(19,24)(22,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151389);