Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #329 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | #1744 | C22 |
| socle | C22 | #1272 |
| prime radical | C23 | #323 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1744 ◁ G | |
| LECS | G ▷ #1744 ▷ C2×C42 ▷ C22 ▷ C1 | |
169 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 169 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18,20,24,23,25,29,31,22,27,30,32,21,26,19,28);(17,19,22,29)(18,24)(20,23,30,21)(25,28)(26,31)(27,32)Polycyclic presentation:
2^9|a^2=d,b^2=f,c^2=fh,d^2=g,e^2=h,g^2=i,b^a=bc,c^a=ce,f^a=fh,c^b=cf,d^b=def,e^b=eh,g^b=gh,d^c=dh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 81);