Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94670 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1402 |
| commutator | #384 | C2×C4 |
| Frattini, cosocle | #1844 | C22 |
| prime radical | C23 | AΣL(1,𝔽16) |
| LCS | G ▷ #384 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1844 ◁ G | |
| LECS | G ▷ #1844 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
164 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5,6,7)(9,10,12,15,20,16,22,24,14,18,11,19,13,17,21,23);(1,2)(3,7)(4,6)(5,8)(9,11,20,21,14,12,13,22)(10,15,18,19)(16,23,17,24)Polycyclic presentation:
2^9|a^2=d,b^2=egi,c^2=fgi,d^2=ghi,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=defghi,e^b=eh,f^b=fh,g^b=gh,d^c=dghi,e^c=eh,f^c=fi,g^c=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1929);