Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(10,17)(11,23,26,40)(12,27)(13,16)(14,29)(18,33,24,37)(19,28)(21,22)(25,30,31,39)(32,35,38,34);(1,2,4,6)(3,7,8,5)(9,10,16,22,15,21,13,17)(11,24,31,32,26,18,25,38)(12,28,14,20,29,19,27,36)(23,34,30,33,40,35,39,37);(1,3,4,8)(2,5,6,7)(9,11,15,26)(10,18,21,24)(12,23,29,40)(13,25,16,31)(14,30,27,39)(17,32,22,38)(19,34,28,35)(20,37,36,33);(9,12,13,27,15,29,16,14)(10,19,17,20,21,28,22,36)(11,23,25,39,26,40,31,30)(18,34,32,37,24,35,38,33)Polycyclic presentation:
2^9|a^2=eh,b^2=eg,c^2=g,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238779, #72239798, #72239829, … |
| maximal normal subgroup of | #72238779, #72239798, #72239829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414814);