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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1427 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26,31,28)(27,32,37,38)(29,33,30,34)(35,39,40,36);(1,3,4,7)(2,5,6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27,31,37)(26,32,28,38)(29,36,30,39)(33,35,34,40);(2,6)(5,8)(25,28,31,26)(27,35,37,40)(29,33,30,34)(32,39,38,36);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29,31,30)(26,33,28,34)(27,36,37,39)(32,35,38,40);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,36)(29,30)(32,35)(33,34)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589374);