Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #333407 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
508 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2,5,7)(3,4,8,6)(9,10,13,23,14,18,15,19)(11,20,22,24,12,16,21,17)(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)(10,16)(12,14)(13,22)(15,21)(17,23)(18,20)(19,24)(25,27,28,35)(26,30,31,33)(29,34,37,36)(32,38,40,39);(9,12,14,11)(10,17,18,24)(13,22,15,21)(16,19,20,23)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,4,5,6)(2,3,7,8)(9,13)(10,18)(11,21)(12,22)(14,15)(17,24)(27,35)(30,33)(34,36)(38,39)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 252863);