Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #333271 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
616 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,16,12,18,13,17)(11,19,22,24,20,15,21,23)(25,26,28,31)(27,33,35,30)(29,32,37,40)(34,39,36,38);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,37)(39,40);(10,16)(13,14)(15,23)(17,18)(19,24)(21,22)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,4,5,6)(2,3,7,8)(10,17)(11,20)(13,14)(15,23)(16,18)(19,24)(26,31)(27,35)(32,40)(34,36)Polycyclic presentation:
2^9|a^2=ef,b^2=ehi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 252790);