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 |
| prime radical | #57178 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,23)(19,21)(25,26)(27,36)(28,38)(29,31)(30,40)(32,35)(33,39)(34,37);(1,2)(3,6)(4,7)(5,8)(9,10,14,15)(11,16,18,12)(13,17,22,23)(19,24,21,20)(25,27)(26,28)(29,40)(30,33)(31,37)(32,38)(34,39)(35,36);(1,3)(2,5)(4,8)(6,7)(25,28)(26,27)(29,37)(30,39)(31,40)(32,36)(33,34)(35,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,22)(23,24)(25,29,32,39)(26,33,35,31)(27,34,38,40)(28,37,36,30);(9,12,14,16)(10,11,15,18)(13,20,22,24)(17,19,23,21)(25,30,32,37)(26,34,35,40)(27,33,38,31)(28,39,36,29);(1,2)(3,5)(4,7)(6,8)(9,13,14,22)(10,17,15,23)(11,19,18,21)(12,20,16,24)(25,31,32,33)(26,29,35,39)(27,37,38,30)(28,40,36,34)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=g,f^2=g,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, 8329031);