Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500378 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,29,31)(27,33,34,40)(28,36,37,32)(30,38,39,35);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,28,35,29,34,37,38)(26,30,32,33,31,39,36,40);(1,3,4,8)(2,5,6,7)(11,17)(12,19)(14,20)(15,22)(26,31)(30,39)(32,36)(33,40);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(27,34)(30,39)(33,40)(35,38);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=gi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6435624);