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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,5,8,7)(9,10)(11,16)(12,22)(13,14)(15,18)(17,21)(19,24)(20,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,3)(2,7)(4,5)(6,8)(9,11)(10,16)(12,23)(13,19)(14,24)(15,21)(17,18)(20,22)(28,36)(31,38)(34,40)(37,39);(9,12,15,13)(10,14,18,22)(11,19,21,23)(16,20,17,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(2,4)(5,7)(9,13,15,12)(10,14,18,22)(11,19,21,23)(16,24,17,20);(1,4,6,2)(3,5,8,7)(9,14)(10,13)(11,20)(12,18)(15,22)(16,23)(17,19)(21,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,5)(2,8)(3,4)(6,7)(9,11)(10,17)(12,19)(13,23)(14,24)(15,21)(16,18)(20,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131530);