Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
304 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 3 generators on 24 points:
(2,5)(3,7)(6,8)(10,17,11,19)(12,13)(14,16)(18,21,20,24)(22,23);(1,2,4,6)(3,8,7,5)(9,10,15,11)(12,21,22,24)(13,19,23,17)(14,20,16,18);(3,7)(5,8)(9,11,15,10)(12,17,22,19)(13,24,23,21)(14,20,16,18)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=i,d^2=f,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=dfi,e^b=eg,f^b=fi,d^c=dg,e^c=ehi,h^c=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72217815 |
| maximal normal subgroup of | #72217815 |
| maximal quotient of | #72214402, #72214416, #72214451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33684);