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 | |
4567 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)(3,5)(4,8)(6,7)(9,10)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(2,7)(3,8)(10,12)(13,21)(15,17)(19,24)(26,31)(29,32)(33,36)(37,39);(1,3,6,8)(2,4,7,5)(11,18)(15,17)(19,24)(22,23)(26,31)(27,34)(33,36)(35,40);(1,4,6,5)(2,3,7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,2)(3,4)(5,8)(6,7)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,5,6,4)(2,8,7,3)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664238);