Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
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 | #1934 |
| prime radical | C23 | #488 |
| 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 | |
215 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,4,6,5,9,13,8)(3,10,11,16,12,7,15,14)(17,18,22,29,24,31,23,30)(19,32,35,45,37,47,36,41)(20,28,21,27,39,46,38,40)(25,34,26,44,43,48,42,33);(2,6,9,8)(7,14,10,16)(17,19,20,33,23,36,38,48,24,37,39,44,22,35,21,34)(18,25,40,47,29,26,28,41,31,43,27,32,30,42,46,45);(1,3,4,11,5,12,13,15)(2,7,8,16,9,10,6,14)(17,19,20,33,23,36,38,48,24,37,39,44,22,35,21,34)(18,26,27,45,29,43,46,47,31,42,40,41,30,25,28,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=d,d^2=gi,e^2=hi,f^2=ghi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^c=egh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47524);