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 | #2537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4286 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)(3,7,5,8)(9,10)(11,17)(12,22)(13,18)(14,15)(16,19)(20,23)(21,24)(25,26)(27,37)(28,34)(29,31)(30,35)(32,36)(33,40)(38,39);(1,3,6,5)(2,7,4,8)(9,11,16,14)(10,17,19,15)(12,21,13,20)(18,23,22,24)(25,27,32,40)(26,33,36,37)(28,38,29,30)(31,39,34,35);(1,3,6,5)(2,8,4,7)(9,12,16,13)(10,18,19,22)(11,20,14,21)(15,23,17,24)(25,28,32,29)(26,34,36,31)(27,38,40,30)(33,35,37,39);(1,4,6,2)(3,7,5,8)(9,13,16,12)(10,18,19,22)(11,21,14,20)(15,23,17,24)(25,29,32,28)(26,34,36,31)(27,30,40,38)(33,35,37,39);(1,5,6,3)(2,7,4,8)(9,14,16,11)(10,17,19,15)(12,21,13,20)(18,24,22,23)(25,30)(26,35)(27,28)(29,40)(31,33)(32,38)(34,37)(36,39);(9,15,16,17)(10,14,19,11)(12,23,13,24)(18,21,22,20)(25,31,32,34)(26,29,36,28)(27,39,40,35)(30,37,38,33)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8585643);