Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | #379 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
796 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 796 conjugacy classes of subgroups — too many to draw. Only its 184 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) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(10,14)(11,15)(12,16)(17,18,19,21,20,22,23,24);(1,2,4,6)(3,7,8,5)(9,10,13,11)(12,15,16,14);(9,11,13,10)(12,14,16,15)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4862);