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 | |
4451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,20,14,16)(12,21,22,18)(13,19,24,23)(25,26)(27,36)(28,30)(29,34)(31,35)(32,39)(33,38)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,20)(12,19,22,23)(13,18,24,21)(25,27)(26,32)(28,37)(29,38)(30,34)(31,39)(33,40)(35,36);(1,3)(2,4)(5,6)(7,8)(10,17)(12,22)(16,20)(19,23)(25,28,31,38)(26,30,35,33)(27,37,39,29)(32,34,36,40);(3,6)(4,8)(9,12,15,22)(10,18,17,21)(11,19,14,23)(13,16,24,20)(25,28)(26,33)(27,37)(29,39)(30,35)(31,38)(32,40)(34,36);(1,4,5,8)(2,6,7,3)(9,13,15,24)(10,19,17,23)(11,21,14,18)(12,20,22,16)(25,29,31,37)(26,34,35,40)(27,38,39,28)(30,36,33,32);(1,2)(3,4)(5,7)(6,8)(9,14,15,11)(10,20,17,16)(12,23,22,19)(13,21,24,18)(25,30,31,33)(26,28,35,38)(27,34,39,40)(29,32,37,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068276);