Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(11,21)(12,26)(13,30)(14,29)(16,34)(17,18)(19,32)(22,38)(23,24)(25,31)(27,37)(28,33)(35,40)(36,39);(1,3,5,4)(2,6,8,7)(9,10)(11,22)(12,27)(13,17)(14,32)(15,20)(16,23)(18,30)(19,29)(21,38)(24,34)(25,40)(26,37)(28,36)(31,35)(33,39);(9,11,15,25)(10,16,20,36)(12,28,30,23)(13,31,26,21)(14,33,29,24)(17,22,37,40)(18,34,27,39)(19,38,32,35);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408130);