Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500596 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 144 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)(11,23,28,25)(12,14)(13,15)(18,34,36,38)(19,37)(21,22)(24,40,33,35)(26,32,27,39)(29,30);(1,2,4,6)(3,7,8,5)(9,10,13,17,16,22,15,21)(11,24,27,36,28,33,26,18)(12,19,14,31,30,37,29,20)(23,34,32,35,25,38,39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,18)(12,25)(13,26)(14,32)(15,27)(16,28)(17,33)(19,35)(20,38)(21,24)(22,36)(23,30)(29,39)(31,34)(37,40);(9,12,13,14,16,30,15,29)(10,19,17,31,22,37,21,20)(11,25,26,32,28,23,27,39)(18,35,33,34,36,40,24,38)Polycyclic presentation:
2^9|a^2=eh,b^2=g,c^2=gh,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239798, #72239811, #72239821, … |
| maximal normal subgroup of | #72239798, #72239811, #72239821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414798);