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 | |
4411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,16,19)(11,20,15,17)(12,14,13,18)(21,23,22,24)(25,26)(27,33)(28,37)(29,32)(30,39)(31,34)(35,36)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,16,15)(10,17,19,20)(12,22,13,21)(14,23,18,24)(25,27,31,29)(26,32,34,33)(28,35,39,38)(30,36,37,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,23,20,24)(25,27,31,29)(26,33,34,32)(28,38,39,35)(30,36,37,40);(1,2,4,5)(3,6,7,8)(9,13,16,12)(10,14,19,18)(11,22,15,21)(17,24,20,23)(25,28)(26,30)(27,35)(29,38)(31,39)(32,36)(33,40)(34,37);(1,2)(3,6)(4,5)(7,8)(9,14)(10,12)(11,23)(13,19)(15,24)(16,18)(17,22)(20,21)(25,29,31,27)(26,32,34,33)(28,35,39,38)(30,40,37,36);(1,3)(2,6)(4,7)(5,8)(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,24,18,23)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081601);