Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9747 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
296 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,7,12,8,13,20,30,5,14,15,31,25,11,6,26)(3,9,17,28,16,21,10,4,19,22,29,23,18,32,27,24);(1,3,15,10,8,18,6,17,5,19,7,27,25,16,20,29)(2,9,26,28,11,21,31,4,14,22,30,23,13,32,12,24);(1,4,20,32,25,28,7,22,5,24,6,21,8,23,15,9)(2,10,12,19,13,29,30,18,14,27,31,3,11,17,26,16)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=egh,d^2=ghi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=dh,f^b=fhi,h^b=hi,d^c=di50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48512);