Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244791 |
| 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 | |
568 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 568 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) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,12,16,13,17,22,15)(11,18,19,23,20,24,21,14)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,36,40,39);(2,5)(6,7)(9,11,13,20)(10,14,17,23)(12,21,22,19)(15,18,16,24)(25,27,29,34)(26,31,30,38)(28,36,37,39)(32,33,35,40);(10,15)(12,22)(14,18)(16,17)(19,21)(23,24)(25,28,29,37)(26,32,30,35)(27,33,34,40)(31,39,38,36);(1,3,4,8)(2,6,5,7)(10,15)(12,22)(14,18)(16,17)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=ei,b^2=hi,c^2=i,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 219356);