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 | |
4462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,20)(11,21,24,17)(12,14,13,18)(15,22,23,19)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2,5,3)(4,6,8,7)(9,11)(10,17)(12,22)(13,19)(14,15)(16,24)(18,23)(20,21)(25,27,31,38)(26,32,36,37)(28,35,39,29)(30,33,34,40);(2,3)(6,7)(9,12,16,13)(10,18,20,14)(11,22,24,19)(15,17,23,21)(25,28,31,39)(26,33,36,40)(27,35,38,29)(30,32,34,37);(2,3)(4,8)(9,13,16,12)(10,14,20,18)(11,22,24,19)(15,17,23,21)(25,27)(26,32)(28,35)(29,39)(30,40)(31,38)(33,34)(36,37);(1,3,5,2)(4,7,8,6)(9,14)(10,13)(11,23)(12,20)(15,24)(16,18)(17,22)(19,21)(25,29,31,35)(26,34,36,30)(27,28,38,39)(32,40,37,33);(1,4)(2,6)(3,7)(5,8)(9,15,16,23)(10,19,20,22)(11,14,24,18)(12,17,13,21)(25,30,31,34)(26,35,36,29)(27,33,38,40)(28,32,39,37)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012475);