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 | #2543 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4275 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,20)(12,14)(13,24)(15,17)(16,21)(18,23)(19,22)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,40,37);(1,3,5,8)(2,6,4,7)(9,11,15,21)(10,16,17,20)(12,19,23,24)(13,18,22,14)(25,27)(26,31)(28,37)(29,39)(30,36)(32,40)(33,38)(34,35);(1,2,5,4)(3,6,8,7)(10,17)(11,21)(12,23)(13,22)(27,36)(28,38)(31,35)(32,39);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,19)(13,20)(14,17)(15,23)(16,22)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(1,4,5,2)(3,7,8,6)(9,13)(10,19)(11,18)(12,20)(14,21)(15,22)(16,23)(17,24)(25,28,30,38)(26,32,34,39)(27,37,36,33)(29,31,40,35);(9,14,15,18)(10,12,17,23)(11,22,21,13)(16,24,20,19)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,35,38,31)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894770);