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 | |
4275 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,19,23)(12,15,20,24)(17,21,18,22)(25,26,31,35)(27,36,37,32)(28,40,34,30)(29,33,39,38);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,23)(17,20)(21,24)(27,37)(29,39)(32,36)(33,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,31,37)(26,32,35,36)(28,33,34,38)(29,30,39,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,28,31,34)(26,30,35,40)(27,38,37,33)(29,36,39,32);(9,12)(10,15)(11,18)(13,20)(14,22)(16,24)(17,19)(21,23)(25,29)(26,33)(27,30)(28,36)(31,39)(32,34)(35,38)(37,40);(9,11)(10,14)(12,17)(13,19)(15,21)(16,23)(18,20)(22,24)(25,30)(26,34)(27,39)(28,35)(29,37)(31,40)(32,38)(33,36)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620517);