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 | |
4668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,8)(6,7)(25,26,31,33)(27,32,34,29)(28,30,37,40)(35,36,39,38);(2,7)(3,8)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,27,31,34)(26,32,33,29)(28,35,37,39)(30,36,40,38);(1,3,6,8)(2,4,7,5)(10,15)(13,22)(16,23)(20,24)(26,33)(27,34)(30,40)(35,39);(1,4,6,5)(2,3,7,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28)(26,30)(27,35)(29,38)(31,37)(32,36)(33,40)(34,39);(1,2)(3,4)(5,8)(6,7)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,29,31,32)(26,27,33,34)(28,36,37,38)(30,39,40,35);(1,5,6,4)(2,8,7,3)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30)(26,28)(27,36)(29,39)(31,40)(32,35)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8809704);