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 | |
4709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,4,5)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40);(1,3)(2,7)(4,6)(5,8)(10,14)(11,17)(16,19)(18,20)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,8)(3,4)(10,14)(11,17)(16,19)(18,20)(26,27)(31,33)(32,34)(38,39);(1,3,6,4)(2,5,8,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,4,6,3)(2,5,8,7)(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,5)(2,4)(3,8)(6,7)(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9976157);