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 | |
4323 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)(4,6)(5,8)(9,10,15,16)(11,12,13,17)(14,22,23,18)(19,24,21,20)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,37)(39,40);(1,3,5,4)(2,6,8,7)(10,16)(12,17)(18,22)(20,24)(26,31)(27,34)(33,36)(35,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,17,16,12)(14,19,23,21)(18,24,22,20)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,2,5,8)(3,6,4,7)(9,12,15,17)(10,11,16,13)(14,20,23,24)(18,19,22,21)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,2)(3,6)(4,7)(5,8)(9,13,15,11)(10,12,16,17)(14,21,23,19)(18,20,22,24);(1,4,5,3)(2,7,8,6)(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=hi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066481);