Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,17,27,34)(7,18,20,28,16,9,14,11)(8,21,23,33,25,13,29,31)(19,22,30,36,32,26,35,24);(2,4)(5,7,11,6,12,16,28,17)(8,19,24,13,25,32,36,21)(9,15,27,20,18,34,10,14)(22,31,23,30,26,33,29,35);(5,8,10,23)(6,13,15,33)(7,19,14,30)(9,26,11,24)(12,25,27,29)(16,32,20,35)(17,21,34,31)(18,22,28,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54716);