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 | |
4491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,20,14,16)(12,21,22,18)(13,19,24,23)(25,26,31,35)(27,36,38,32)(28,33,37,30)(29,34,39,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,20)(12,19,22,23)(13,18,24,21)(25,27,31,38)(26,32,35,36)(28,29,37,39)(30,34,33,40);(1,3)(2,4)(5,6)(7,8)(10,17)(12,22)(16,20)(19,23)(25,28,31,37)(26,33,35,30)(27,29,38,39)(32,40,36,34);(3,6)(4,8)(9,12,15,22)(10,18,17,21)(11,19,14,23)(13,16,24,20)(25,28)(26,30)(27,29)(31,37)(32,34)(33,35)(36,40)(38,39);(1,4,5,8)(2,6,7,3)(9,13,15,24)(10,19,17,23)(11,21,14,18)(12,20,22,16)(25,29)(26,34)(27,37)(28,38)(30,36)(31,39)(32,33)(35,40);(1,2)(3,4)(5,7)(6,8)(9,14,15,11)(10,20,17,16)(12,23,22,19)(13,21,24,18)(25,30)(26,28)(27,34)(29,32)(31,33)(35,37)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=hi,f^2=h,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, 9068893);