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 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,20,14)(12,18,22,23)(13,19,21,24)(25,26)(27,36)(28,30)(29,34)(31,35)(32,39)(33,38)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,20)(10,16,17,14)(12,21,22,13)(18,24,23,19)(25,27)(26,32)(28,37)(29,38)(30,34)(31,39)(33,40)(35,36);(1,3)(2,5)(4,8)(6,7)(10,17)(11,20)(12,22)(19,24)(25,28,31,38)(26,30,35,33)(27,37,39,29)(32,34,36,40);(3,8)(5,6)(9,12)(10,18)(11,21)(13,20)(14,19)(15,22)(16,24)(17,23)(25,28)(26,33)(27,37)(29,39)(30,35)(31,38)(32,40)(34,36);(1,3)(2,6)(4,8)(5,7)(9,13,15,21)(10,19,17,24)(11,22,20,12)(14,18,16,23)(25,29,31,37)(26,34,35,40)(27,38,39,28)(30,36,33,32);(1,2)(3,5)(4,7)(6,8)(9,14,15,16)(10,20,17,11)(12,19,22,24)(13,23,21,18)(25,30,31,33)(26,28,35,38)(27,34,39,40)(29,32,37,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,e^2=gh,f^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, 9032880);