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 | |
4334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,23)(19,21)(25,26,31,35)(27,36,38,32)(28,39,30,33)(29,40,37,34);(1,2)(3,6)(4,7)(5,8)(9,10,14,15)(11,16,18,12)(13,17,22,23)(19,24,21,20)(25,27,31,38)(26,32,35,36)(28,37,30,29)(33,34,39,40);(1,3)(2,5)(4,8)(6,7)(25,28,31,30)(26,33,35,39)(27,37,38,29)(32,34,36,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,22)(23,24)(25,26)(27,36)(28,33)(29,34)(30,39)(31,35)(32,38)(37,40);(9,12,14,16)(10,11,15,18)(13,20,22,24)(17,19,23,21)(25,29)(26,34)(27,28)(30,38)(31,37)(32,39)(33,36)(35,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,22)(10,17,15,23)(11,19,18,21)(12,20,16,24)(25,30,31,28)(26,33,35,39)(27,29,38,37)(32,34,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=gh,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, 8329404);