Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144895 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2041 |
| commutator, Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
430 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2)(3,6)(4,7)(5,8)(9,10,16,22)(11,23,27,35)(12,17,15,19)(13,29,31,21)(14,20,28,39)(18,32,38,26)(24,34,30,36)(25,37,33,40);(1,2)(3,7)(4,6)(5,8)(10,17)(11,24)(12,15)(13,14)(18,35)(19,22)(20,39)(23,38)(25,26)(27,30)(28,31)(32,33)(34,37)(36,40);(2,6)(3,4)(7,8)(9,11,12,25,16,27,15,33)(10,18,19,36,22,38,17,34)(13,30,28,26,31,24,14,32)(20,37,21,23,39,40,29,35)Polycyclic presentation:
2^9|a^2=i,c^2=d,d^2=i,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,e^c=egi,f^c=fg,g^c=gi,h^c=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52092);