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 | #2537 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
580 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,4)(5,7)(6,8)(9,10,13,16,12,17,21,15)(11,18,20,23,19,14,22,24)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,20)(15,23)(16,24)(17,18)(21,22)(25,27)(26,30)(28,34)(29,33)(31,38)(32,40)(35,36)(37,39);(1,4,5,6)(2,3,7,8)(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(2,7)(3,8)(10,16)(11,19)(13,21)(14,23)(15,17)(18,24)(26,31)(30,38)(32,39)(37,40)Polycyclic presentation:
2^9|a^2=fi,c^2=eh,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, 250639);