Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(26,31)(32,40)(33,36)(34,39);(1,3,5,4)(2,6,7,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,26)(27,33)(28,36)(29,31)(30,39)(32,35)(34,38)(37,40);(1,4,5,3)(2,6,7,8)(25,27,29,28)(26,32,31,40)(30,37,38,35)(33,34,36,39);(10,15)(14,17)(16,23)(22,24)(25,28,29,27)(26,33,31,36)(30,37,38,35)(32,34,40,39);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(26,34)(30,38)(31,39)(32,36)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6601126);