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 | |
4569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10,15,17)(11,12,19,13)(14,16,23,24)(18,20,22,21)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(1,3)(2,7)(4,5)(6,8)(9,11)(10,12)(13,17)(14,22)(15,19)(16,21)(18,23)(20,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(9,12,15,13)(10,11,17,19)(14,20,23,21)(16,18,24,22)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(2,4)(5,7)(9,13,15,12)(10,11,17,19)(14,20,23,21)(16,22,24,18)(25,26)(27,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(1,4,6,2)(3,5,8,7)(9,14)(10,16)(11,18)(12,20)(13,21)(15,23)(17,24)(19,22)(25,28,30,27)(26,31,33,34)(29,35,38,36)(32,40,37,39);(1,5)(2,8)(3,4)(6,7)(9,11)(10,13)(12,17)(14,22)(15,19)(16,20)(18,23)(21,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131486);