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, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,12,15,23,25)(7,18,20,35,22,36,34,31)(8,11,17,9,24,14,28,27)(13,29,30,19,32,21,33,26);(2,4)(5,7,11,21,12,22,14,29)(6,13,17,31,15,32,28,35)(8,18,25,30,24,36,16,33)(9,26,10,20,27,19,23,34);(5,8,23,17)(6,14,16,27)(7,18,34,31)(9,15,11,25)(10,28,12,24)(13,29,33,19)(20,35,22,36)(21,30,26,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,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.
| maximal subgroup of | #72311927, #72311949, #72311961, … |
| maximal normal subgroup of | #72311927, #72311949, #72311961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54717);