Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140140 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1925 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9545 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
239 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 239 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12,8,13,26,23,9,14,25,11,5,4,6,24)(3,15,19,31,17,16,18,30,21,22,27,32,20,10,28,29);(1,3,6,18,5,20,25,19,9,21,26,28,8,17,7,27)(2,10,24,31,4,22,11,29,14,16,23,32,13,15,12,30);(1,4,5,14,9,13,8,2)(3,16,17,22,21,10,20,15)(6,12,25,24,26,11,7,23)(18,31,27,30,28,32,19,29)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fi,e^c=egi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47336);