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 | |
4281 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)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(2,7)(3,8)(9,10,14,12)(11,15,18,19)(13,16,21,20)(17,22,23,24)(28,36)(31,38)(34,40)(37,39);(1,3,6,8)(2,4,7,5)(10,12)(13,21)(15,19)(17,23)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,6,5)(2,3,7,8)(9,11,14,18)(10,15,12,19)(13,17,21,23)(16,22,20,24);(1,2)(3,4)(5,8)(6,7)(9,12,14,10)(11,15,18,19)(13,20,21,16)(17,22,23,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,5,6,4)(2,8,7,3)(9,13)(10,16)(11,17)(12,20)(14,21)(15,22)(18,23)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8870831);