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 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(10,14)(15,21)(16,19)(22,24)(25,26,29,31)(27,33,35,30)(28,32,37,39)(34,40,36,38);(2,6)(3,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,29,35)(26,30,31,33)(28,34,37,36)(32,38,39,40);(1,3,5,7)(2,4,6,8)(11,17)(15,21)(18,23)(22,24)(26,31)(30,33)(32,39)(38,40);(1,4)(2,7)(3,6)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,26)(27,30)(28,32)(29,31)(33,35)(34,38)(36,40)(37,39);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442085);