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 | |
4409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,20,22,16)(12,23,19,14)(13,18,24,21)(25,26,30,35)(27,31,34,29)(28,33,39,32)(36,38,37,40);(1,2)(3,4)(5,7)(6,8)(9,11,15,22)(10,16,17,20)(12,21,19,18)(13,23,24,14)(25,27,30,34)(26,31,35,29)(28,37,39,36)(32,38,33,40);(1,3,5,8)(2,6,7,4)(10,17)(12,19)(16,20)(18,21)(25,28,30,39)(26,32,35,33)(27,36,34,37)(29,38,31,40);(3,8)(4,6)(9,12,15,19)(10,14,17,23)(11,21,22,18)(13,20,24,16)(25,28)(26,33)(27,37)(29,38)(30,39)(31,40)(32,35)(34,36);(1,4)(2,3)(5,6)(7,8)(9,13,15,24)(10,18,17,21)(11,14,22,23)(12,16,19,20)(25,27)(26,31)(28,36)(29,35)(30,34)(32,40)(33,38)(37,39);(1,2)(3,6)(4,8)(5,7)(9,14)(10,19)(11,13)(12,17)(15,23)(16,18)(20,21)(22,24)(25,29,30,31)(26,34,35,27)(28,38,39,40)(32,37,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9045647);