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 | |
4461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,15,16)(11,17,21,24)(12,14,20,18)(13,19,22,23)(25,26,31,35)(27,34,37,29)(28,38,39,32)(30,33,40,36);(1,2)(3,5)(4,6)(7,8)(10,16)(11,21)(12,20)(19,23)(25,27)(26,29)(28,36)(30,32)(31,37)(33,39)(34,35)(38,40);(9,11)(10,17)(12,22)(13,20)(14,23)(15,21)(16,24)(18,19)(25,28,31,39)(26,32,35,38)(27,36,37,33)(29,30,34,40);(1,3,4,7)(2,5,6,8)(9,12,15,20)(10,18,16,14)(11,22,21,13)(17,19,24,23)(25,29,31,34)(26,27,35,37)(28,30,39,40)(32,36,38,33);(9,13,15,22)(10,19,16,23)(11,20,21,12)(14,17,18,24)(25,30)(26,33)(27,32)(28,34)(29,39)(31,40)(35,36)(37,38);(9,14,15,18)(10,20,16,12)(11,23,21,19)(13,24,22,17)(25,27)(26,34)(28,36)(29,35)(30,38)(31,37)(32,40)(33,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565645);