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 | #2464 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4262 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)(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26)(27,31)(28,32)(29,38)(30,33)(34,36)(35,39)(37,40);(1,2,6,5)(3,8,7,4)(10,15)(12,17)(16,21)(20,22)(25,27,30,37)(26,31,33,40)(28,35,38,34)(29,36,32,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(16,21)(19,24)(25,28,30,38)(26,29,33,32)(27,34,37,35)(31,39,40,36);(1,2,6,5)(3,4,7,8)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,28)(26,32)(27,35)(29,33)(30,38)(31,39)(34,37)(36,40);(1,4,6,8)(2,7,5,3)(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,27)(26,31)(28,34)(29,39)(30,37)(32,36)(33,40)(35,38);(1,5,6,2)(3,8,7,4)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,29,30,32)(26,28,33,38)(27,36,37,39)(31,35,40,34)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9071140);