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 | #48669 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,14,12)(11,15,19,16)(13,20,21,17)(18,23,24,22)(25,26)(27,37)(28,31)(29,38)(30,34)(32,36)(33,40)(35,39);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27)(26,33)(28,38)(29,34)(30,35)(31,39)(32,40)(36,37);(9,11,14,19)(10,16,12,15)(13,18,21,24)(17,23,20,22)(25,28,32,30)(26,34,36,31)(27,38,40,35)(29,37,39,33);(2,4)(7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,29,32,39)(26,35,36,38)(27,34,40,31)(28,33,30,37);(1,4,6,2)(3,8,5,7)(9,12,14,10)(11,15,19,16)(13,17,21,20)(18,23,24,22)(25,30,32,28)(26,34,36,31)(27,38,40,35)(29,33,39,37);(1,5,6,3)(2,7,4,8)(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,31,32,34)(26,30,36,28)(27,39,40,29)(33,35,37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=hi,e^2=gi,f^2=gi,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, 10123814);