Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #334823 |
| 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 | |
549 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,4,7,6)(9,10,13,17,14,18,11,15)(12,20,19,23,21,24,22,16)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(1,3,5,7)(2,6,8,4)(9,11,14,13)(10,15,18,17)(12,19,21,22)(16,20,23,24)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(1,4,5,6)(2,7,8,3)(10,15)(11,13)(16,20)(17,18)(19,22)(23,24)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(2,8)(3,7)(9,12,14,21)(10,16,18,23)(11,19,13,22)(15,20,17,24)Polycyclic presentation:
2^9|a^2=efi,b^2=ehi,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,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, 252520);