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 | |
4198 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)(3,7)(4,6)(5,8)(10,13)(15,18)(16,21)(22,24)(25,26,32,36)(27,37,39,33)(28,30,34,40)(29,31,35,38);(1,2,4,6)(3,5,8,7)(12,20)(16,21)(17,23)(22,24)(25,27,32,39)(26,33,36,37)(28,38,34,31)(29,30,35,40);(1,2)(3,5)(4,6)(7,8)(9,10,14,13)(11,15,19,18)(12,16,20,21)(17,22,23,24)(25,28,32,34)(26,30,36,40)(27,38,39,31)(29,33,35,37);(1,3)(2,5)(4,8)(6,7)(9,11,14,19)(10,15,13,18)(12,17,20,23)(16,22,21,24)(25,29)(26,31)(27,30)(28,33)(32,35)(34,37)(36,38)(39,40);(9,12,14,20)(10,16,13,21)(11,17,19,23)(15,22,18,24)(25,30)(26,34)(27,35)(28,36)(29,39)(31,33)(32,40)(37,38);(9,13,14,10)(11,18,19,15)(12,21,20,16)(17,24,23,22)(25,31)(26,35)(27,28)(29,36)(30,37)(32,38)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9143664);