Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #333271 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
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 | |
500 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 500 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:
(1,2,5,7)(3,4,8,6)(9,10,13,18,14,19,23,17)(11,15,20,16,12,21,22,24)(26,30)(27,33)(32,37)(35,38);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,15,19,21)(13,20,23,22)(16,17,24,18)(25,26,28,30)(27,34,33,40)(29,37,36,32)(31,38,39,35);(9,12,14,11)(10,16,19,24)(13,20,23,22)(15,17,21,18)(25,27,28,33)(26,31,30,39)(29,38,36,35)(32,40,37,34);(1,4,5,6)(2,3,7,8)(10,17)(13,23)(15,24)(16,21)(18,19)(20,22)(26,30)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=hi,d^2=ei,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, 252791);