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 | |
4432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,12)(11,16,19,15)(13,20,21,17)(18,22,24,23)(25,26,31,33)(27,32,30,40)(28,36,38,29)(34,39,37,35);(1,3)(2,6)(4,7)(5,8)(9,11,14,19)(10,15,12,16)(13,18,21,24)(17,22,20,23)(25,27,31,30)(26,32,33,40)(28,37,38,34)(29,39,36,35);(1,3)(2,7)(4,6)(5,8)(9,11,14,19)(10,15,12,16)(13,18,21,24)(17,22,20,23)(25,28,31,38)(26,29,33,36)(27,34,30,37)(32,35,40,39);(1,2,5,4)(3,7,8,6)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,29)(27,34)(30,37)(31,38)(32,35)(33,36)(39,40);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,15,19,16)(13,17,21,20)(18,23,24,22)(25,29)(26,28)(27,35)(30,39)(31,36)(32,34)(33,38)(37,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,30,31,27)(26,32,33,40)(28,37,38,34)(29,35,36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947910);