Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136222 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
365 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(5,8)(9,10,14,21,16,22,15,33)(11,23,25,37,27,34,26,38)(12,28,31,20,13,17,29,19)(18,30,35,24,36,32,39,40);(1,2)(3,5)(4,6)(7,8)(10,17,22,28)(12,29)(13,31)(14,15)(18,34,36,23)(19,21,20,33)(24,30)(25,26)(32,40)(35,37,39,38);(1,3)(2,5)(4,7)(6,8)(9,11,16,27)(10,18,22,36)(12,30,13,32)(14,26,15,25)(17,34,28,23)(19,37,20,38)(21,39,33,35)(24,31,40,29)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=i,d^2=g,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43420);