Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1942 |
| prime radical | C23 | #499 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(17,18,22,26,23,28,37,27)(19,24,30,38,31,39,21,36)(20,32,34,45,35,47,48,46)(25,29,40,43,42,44,41,33);(1,3,5,11,6,12,4,14)(2,7,8,15,9,16,13,10)(17,19,22,30,23,31,37,21)(18,24,26,38,28,39,27,36)(20,29,34,43,35,44,48,33)(25,32,40,45,42,47,41,46);(17,20,31,44,22,34,21,33,23,35,19,29,37,48,30,43)(18,25,38,45,27,41,24,32,28,42,36,46,26,40,39,47)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ehi,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47680);