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 | |
4426 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)(27,29)(28,36)(30,32)(31,34)(33,39)(35,40)(37,38);(1,3,6,5)(2,7,4,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,30,34)(26,31,32,29)(28,37,39,35)(33,38,36,40);(1,3,6,5)(2,8,4,7)(10,14)(12,21)(15,17)(18,22)(26,32)(29,31)(33,36)(38,40);(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,28,30,39)(26,33,32,36)(27,35,34,37)(29,38,31,40);(1,4,6,2)(3,7,5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,26)(27,31)(28,36)(29,34)(30,32)(33,39)(35,38)(37,40);(1,5,6,3)(2,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,29,30,31)(26,34,32,27)(28,38,39,40)(33,37,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=gh,f^2=h,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, 8948153);