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 | |
4501 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,23)(20,24)(27,33)(31,34)(35,39)(38,40);(1,3,6,7)(2,4,5,8)(10,15)(11,18)(12,21)(20,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,4,6,8)(2,7,5,3)(10,15)(11,18)(12,21)(20,23)(26,30)(31,34)(32,36)(38,40);(1,2,6,5)(3,8,7,4)(9,11,14,18)(10,16,15,22)(12,19,21,24)(13,20,17,23)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,21)(10,17,15,13)(11,19,18,24)(16,23,22,20)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(1,5,6,2)(3,8,7,4)(9,13,14,17)(10,12,15,21)(11,20,18,23)(16,19,22,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=h,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, 9945663);