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 | |
4334 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)(3,7)(4,5)(6,8)(10,15)(11,17)(18,23)(19,24)(25,26,31,34)(27,29,33,39)(28,30,38,32)(35,40,36,37);(1,2,5,4)(3,6,8,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(25,27,31,33)(26,29,34,39)(28,36,38,35)(30,37,32,40);(1,3)(2,6)(4,7)(5,8)(25,28,31,38)(26,32,34,30)(27,35,33,36)(29,37,39,40);(1,2,5,4)(3,7,8,6)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20)(25,28)(26,30)(27,36)(29,37)(31,38)(32,34)(33,35)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,29,31,39)(26,33,34,27)(28,37,38,40)(30,35,32,36);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,30)(26,28)(27,37)(29,36)(31,32)(33,40)(34,38)(35,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592731);