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 | |
4383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(9,10,15,17)(11,16,20,14)(12,18,22,24)(13,23,21,19);(1,3,6,5)(2,7,8,4)(9,11)(10,16)(12,13)(14,17)(15,20)(18,23)(19,24)(21,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,6,5)(2,7,8,4)(10,17)(11,20)(12,22)(19,23)(26,31)(27,35)(33,37)(34,36);(2,8)(3,5)(9,12)(10,18)(11,21)(13,20)(14,23)(15,22)(16,19)(17,24)(25,27,30,35)(26,32,31,39)(28,36,38,34)(29,33,40,37);(1,4)(2,3)(5,8)(6,7)(9,13,15,21)(10,19,17,23)(11,12,20,22)(14,18,16,24)(25,28)(26,33)(27,36)(29,39)(30,38)(31,37)(32,40)(34,35);(1,5,6,3)(2,7,8,4)(9,14,15,16)(10,11,17,20)(12,19,22,23)(13,24,21,18)(25,29,30,40)(26,34,31,36)(27,37,35,33)(28,32,38,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8632005);