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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4167 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,17)(13,18)(15,16)(19,24)(20,22)(21,23)(26,31)(27,34)(33,40)(35,36);(2,5)(6,7)(9,11,15,19)(10,14,16,24)(12,20,21,13)(17,22,23,18)(25,26,30,31)(27,32,34,29)(28,33,38,40)(35,37,36,39);(1,2)(3,6)(4,5)(7,8)(10,16)(11,19)(12,21)(18,22);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,20)(13,19)(14,22)(15,21)(16,23)(18,24)(25,27)(26,32)(28,36)(29,31)(30,34)(33,39)(35,38)(37,40);(9,13,15,20)(10,18,16,22)(11,21,19,12)(14,23,24,17)(25,28,30,38)(26,33,31,40)(27,35,34,36)(29,39,32,37);(9,14)(10,19)(11,16)(12,22)(13,17)(15,24)(18,21)(20,23)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,36,40,35)Polycyclic presentation:
2^9|b^2=gi,d^2=hi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10123954);