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 | |
4587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,17)(13,18)(15,16)(19,24)(20,22)(21,23)(25,26)(27,36)(28,33)(29,30)(31,35)(32,39)(34,38)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,19)(10,14,16,24)(12,20,21,13)(17,22,23,18)(25,27)(26,32)(28,37)(29,38)(30,33)(31,39)(34,40)(35,36);(1,3)(2,5)(4,8)(6,7)(10,16)(11,19)(12,21)(18,22)(25,28,31,38)(26,33,35,34)(27,37,39,29)(30,36,40,32);(3,8)(5,6)(9,12)(10,17)(11,20)(13,19)(14,22)(15,21)(16,23)(18,24)(25,28)(26,34)(27,37)(29,39)(30,36)(31,38)(32,40)(33,35);(1,3)(2,6)(4,8)(5,7)(9,13,15,20)(10,18,16,22)(11,21,19,12)(14,23,24,17)(25,29,31,37)(26,30,35,40)(27,38,39,28)(32,33,36,34);(1,2)(3,5)(4,7)(6,8)(9,14)(10,19)(11,16)(12,22)(13,17)(15,24)(18,21)(20,23)(25,30)(26,29)(27,33)(28,36)(31,40)(32,38)(34,39)(35,37)Polycyclic presentation:
2^9|b^2=h,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9030814);