Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93852 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7089 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2150 | C22 |
| prime radical | C23 | #360 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ #2150 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,8,9,25,13,28,12,27)(3,14)(4,19)(5,23)(6,22)(10,29,11,26,24,15,20,30)(16,31)(17,32)(18,21);(1,2,6,9,7,13,22,12)(3,15,17,26,18,29,31,30)(4,20,19,10,23,11,5,24)(8,21,27,32,28,14,25,16)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=fh,d^2=ghi,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=defhi,e^b=eh,f^b=fhi,h^b=hi,d^c=dghi,e^c=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1049);