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 | |
4553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,21)(16,19)(22,23)(25,26)(27,30)(28,39)(29,31)(32,37)(33,36)(34,38)(35,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,27,32,36)(26,33,37,30)(28,31,38,35)(29,39,40,34);(1,3)(2,5)(4,8)(6,7)(25,28)(26,34)(27,31)(29,33)(30,40)(32,38)(35,36)(37,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,29,32,40)(26,35,37,31)(27,34,36,39)(28,33,38,30);(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,30,32,33)(26,36,37,27)(28,40,38,29)(31,34,35,39);(1,2)(3,5)(4,7)(6,8)(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23)(25,31,32,35)(26,29,37,40)(27,38,36,28)(30,34,33,39)Polycyclic presentation:
2^9|b^2=h,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, 8329029);