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 | |
4266 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)(11,13)(12,16)(14,18)(15,22)(17,23)(19,24)(20,21)(25,26,32,30)(27,36,38,33)(28,34,29,39)(31,35,37,40);(1,2,5,4)(3,6,8,7)(9,11,14,17)(10,13,18,23)(12,20,22,19)(15,24,16,21)(25,27)(26,33)(28,31)(29,37)(30,36)(32,38)(34,40)(35,39);(1,2,5,4)(3,7,8,6)(9,12,14,22)(10,15,18,16)(11,19,17,20)(13,21,23,24)(25,28,32,29)(26,34,30,39)(27,31,38,37)(33,40,36,35);(1,3,5,8)(2,6,4,7)(9,12)(10,16)(11,20)(13,21)(14,22)(15,18)(17,19)(23,24)(25,29,32,28)(26,34,30,39)(27,37,38,31)(33,40,36,35);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,19)(14,17)(15,21)(16,24)(18,23)(20,22)(25,30,32,26)(27,36,38,33)(28,39,29,34)(31,35,37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,18)(12,21)(14,23)(15,20)(16,19)(22,24)(25,31,32,37)(26,35,30,40)(27,28,38,29)(33,39,36,34)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gi,d^2=g,e^2=ghi,f^2=g,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, 10066413);