Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #121 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×D4 | M16 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
359 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,6,5,8,3,7)(9,10,13,18,14,17,27,30,15,19,28,33,12,26,21,36)(11,20,24,35,25,31,38,39,22,34,37,40,23,16,29,32);(2,6)(3,4)(7,8)(9,11,21,29,12,23,28,37,15,22,27,38,14,25,13,24)(10,16,30,35,26,34,18,32,19,31,36,40,17,20,33,39);(2,6)(3,4)(7,8)(11,22)(16,31)(20,34)(23,25)(24,37)(29,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ci,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43849);