Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C42 |
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 | |
877 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)(5,6,9,15)(7,17,21,12)(8,10,24,27)(11,16,26,13)(14,23,25,28)(18,33,31,20)(19,30,22,32)(29,36,35,34);(2,4)(5,7)(6,12)(8,18)(9,21,26,19)(10,20,28,36)(11,22)(13,17,15,32)(14,31,24,29)(16,30)(23,34)(25,35)(27,33);(5,8,23,16,11,25,27,6)(7,18,34,30,22,35,33,12)(9,14,10,13,26,24,28,15)(17,19,29,36,32,21,31,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^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 | #72311838, #72311842, #72311871, … |
| maximal normal subgroup of | #72311838, #72311842, #72311871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54713);