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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(10,15)(11,18)(17,24)(19,20)(25,26,31,36)(27,34,38,32)(28,39,35,30)(29,37,40,33);(2,6)(3,7)(9,10,14,15)(11,16,18,13)(12,17,22,24)(19,21,20,23)(25,27,31,38)(26,32,36,34)(28,29,35,40)(30,37,39,33);(1,2,5,6)(3,4,7,8)(25,28,31,35)(26,30,36,39)(27,29,38,40)(32,37,34,33);(2,6)(3,7)(9,11)(10,16)(12,20)(13,15)(14,18)(17,23)(19,22)(21,24)(25,29,31,40)(26,33,36,37)(27,35,38,28)(30,34,39,32);(1,3)(2,4)(5,7)(6,8)(9,12,14,22)(10,17,15,24)(11,19,18,20)(13,23,16,21)(25,27)(26,34)(28,29)(30,33)(31,38)(32,36)(35,40)(37,39);(1,4)(2,7)(3,6)(5,8)(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,20,24,19)(25,30)(26,35)(27,37)(28,36)(29,34)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123710);