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 |
| prime radical | #46375 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(25,26,31,34)(27,33,29,32)(28,37,39,30)(35,36,38,40);(1,3,6,5)(2,7,4,8)(9,10,14,15)(11,16,18,23)(12,13,17,22)(19,24,21,20)(25,27,31,29)(26,32,34,33)(28,38,39,35)(30,40,37,36);(1,3,6,5)(2,8,4,7)(10,15)(11,18)(13,22)(19,21)(25,28)(26,30)(27,35)(29,38)(31,39)(32,36)(33,40)(34,37);(9,11)(10,16)(12,21)(13,20)(14,18)(15,23)(17,19)(22,24)(25,29,31,27)(26,33,34,32)(28,38,39,35)(30,40,37,36);(1,4,6,2)(3,7,5,8)(9,12,14,17)(10,13,15,22)(11,19,18,21)(16,24,23,20)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39);(1,5,6,3)(2,7,4,8)(9,13)(10,17)(11,20)(12,15)(14,22)(16,19)(18,24)(21,23)(25,26)(27,32)(28,30)(29,33)(31,34)(35,36)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8826665);