Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,34)(28,33)(29,38)(30,36)(31,32)(35,39)(37,40);(1,2)(3,7)(4,5)(6,8)(10,11)(12,19)(16,18)(20,24)(26,32)(29,39)(33,34)(36,37);(2,5)(6,7)(13,21)(16,18)(20,24)(22,23)(25,27,31,28)(26,33,32,34)(29,36,39,37)(30,38,40,35);(1,3,4,8)(2,6,5,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,27)(26,34,32,33)(29,36,39,37)(30,38,40,35);(1,3)(2,6)(4,8)(5,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,39)(26,35,32,38)(27,36,28,37)(30,34,40,33);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,40)(26,36,32,37)(27,35,28,38)(29,34,39,33)Polycyclic presentation:
2^9|c^2=h,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8298119);