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 | |
4504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(26,31)(29,39)(32,38)(33,40);(1,2,6,5)(3,8,7,4)(9,10,13,16)(11,14,19,17)(12,15,21,20)(18,22,24,23)(27,35)(28,37)(32,38)(33,40);(1,3)(2,4)(5,8)(6,7)(25,26,30,31)(27,32,35,38)(28,33,37,40)(29,34,39,36);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,36)(29,33)(30,35)(31,38)(34,37)(39,40);(1,4,6,8)(2,7,5,3)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,28,30,37)(26,33,31,40)(27,36,35,34)(29,38,39,32);(1,5,6,2)(3,8,7,4)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29)(26,34)(27,33)(28,38)(30,39)(31,36)(32,37)(35,40)Polycyclic presentation:
2^9|b^2=g,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000423);