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 | |
4371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,16)(11,14,18,22)(12,15,20,19)(17,21,24,23)(25,26,31,32)(27,34,29,33)(28,37,38,30)(35,36,39,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(26,32)(28,38)(33,34)(35,39);(1,3,5,6)(2,4,7,8)(25,27,31,29)(26,33,32,34)(28,35,38,39)(30,36,37,40);(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,23)(18,22)(21,24)(25,28,31,38)(26,30,32,37)(27,35,29,39)(33,36,34,40);(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,29,31,27)(26,34,32,33)(28,39,38,35)(30,40,37,36);(1,4)(2,6)(3,7)(5,8)(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,21,22,23)(25,30)(26,28)(27,36)(29,40)(31,37)(32,38)(33,35)(34,39)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332081);