Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #278923 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
538 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,12,15)(11,17,20,24)(13,22,19,16)(14,23,18,21)(25,26,28,31)(27,33,35,30)(29,32,37,39)(34,38,36,40);(1,3,5,4)(2,6,8,7)(9,10)(11,18)(12,15)(13,16)(14,20)(17,23)(19,22)(21,24)(25,27,29,36,28,35,37,34)(26,30,32,38,31,33,39,40);(1,3,5,4)(2,7,8,6)(9,11,19,23,12,20,13,21)(10,14,22,24,15,18,16,17)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(1,4,5,3)(2,7,8,6)(10,15)(14,18)(16,22)(17,24)(26,31)(30,33)(32,39)(38,40)Polycyclic presentation:
2^9|a^2=e,b^2=egi,c^2=ef,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 232837);