Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7535 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1409 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1848 | C22 |
| socle | C2 | #7355 |
| prime radical | C23 | #355 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1848 ◁ G | |
| LECS | G ▷ #1848 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,7,13,3,15,6,14,18,29,8,12,17,31)(4,19,16,9,22,10,20,25,21,27,32,26,23,28,24,30);(1,3,6,17)(2,9)(4,20,21,24)(5,7,18,8)(10,11)(12,30)(13,25)(14,26)(15,27)(16,22,32,23)(19,31)(28,29)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=ghi,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cghi,d^b=degi,e^b=ehi,g^b=gi,d^c=dgh,e^d=ehi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1770);