Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7533 |
| 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 | |
127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,7,11,3,13,6,12,16,29,19,30,15,28)(4,17,20,8,22,24,23,26,21,32,14,27,31,9,18,25);(1,3,6,15)(2,8)(4,18,21,23)(5,7,16,19)(9,10)(11,26)(12,27)(13,17)(14,22,20,31)(24,29)(25,30)(28,32)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=gh,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=cgh,d^b=deg,e^b=ehi,g^b=gi,d^c=dgh,e^d=ehi,g^d=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1769);