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 | |
4499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(25,26)(27,36)(28,29)(30,37)(31,33)(32,38)(34,39)(35,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,31,38)(26,32,33,36)(28,35,39,37)(29,30,34,40);(1,3,5,8)(2,6,7,4)(10,14)(12,21)(15,17)(18,22)(26,33)(29,34)(30,40)(32,36);(2,7)(4,6)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,28,31,39)(26,34,33,29)(27,37,38,35)(30,36,40,32);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29,31,34)(26,28,33,39)(27,30,38,40)(32,35,36,37);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,30)(26,35)(27,34)(28,36)(29,38)(31,40)(32,39)(33,37)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947803);