Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7519 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1402 |
| commutator | C4⋊Q8 | C2×C8 |
| Frattini, cosocle | #1956 | C22 |
| prime radical | C23 | AΣL(1,𝔽16) |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1956 ◁ G | |
| LECS | G ▷ #1956 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
237 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 28 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:
(1,2,3,5,4,6,7,8)(9,10)(11,17,20,16)(12,21,22,15)(13,18)(14,19,24,23);(9,11,13,20)(10,14,18,21)(12,22,23,19)(15,16,24,17)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=deh,e^b=ehi,g^b=ghi,d^c=dgh,e^c=ehi,g^c=gi,e^d=ehi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1737);