Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #331791 |
| 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 | |
498 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 498 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)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26,29,31)(27,30,34,38)(28,32,35,39)(33,37,36,40);(1,3,5,8)(2,6,7,4)(10,14)(15,23)(16,21)(17,24)(27,33)(28,35)(30,37)(32,39)(34,36)(38,40);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(25,27,29,34)(26,30,31,38)(28,36,35,33)(32,40,39,37);(1,4,5,6)(2,3,7,8)(10,16)(13,20)(14,21)(15,24)(17,23)(19,22)(26,31)(30,38)(32,39)(37,40)Polycyclic presentation:
2^9|a^2=ei,b^2=eh,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 248865);