Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 166 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)(3,5)(4,7)(6,8)(9,10,13,20,16,21,15,30)(11,22,24,32,27,38,26,40)(12,18,28,35,29,36,14,19)(17,31,33,39,34,25,37,23);(2,5,7,6)(9,11,14,25,15,26,29,39,16,27,28,31,13,24,12,23)(10,17,18,32,20,33,35,38,21,34,36,40,30,37,19,22);(9,12,13,28,16,29,15,14)(10,18,20,35,21,36,30,19)(11,23,24,31,27,39,26,25)(17,32,33,38,34,40,37,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,h^a=hi176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35729);