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 | |
4407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,7)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(25,26)(27,36)(28,38)(29,31)(30,40)(32,35)(33,39)(34,37);(1,2)(3,6)(4,5)(7,8)(10,16)(11,12)(18,21)(19,20)(25,27)(26,28)(29,40)(30,33)(31,37)(32,38)(34,39)(35,36);(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,28)(26,27)(29,37)(30,39)(31,40)(32,36)(33,34)(35,38);(1,2)(3,7)(4,5)(6,8)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,29,32,39)(26,33,35,31)(27,34,38,40)(28,37,36,30);(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,30,32,37)(26,34,35,40)(27,33,38,31)(28,39,36,29);(1,3,4,8)(2,6,5,7)(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,31,32,33)(26,29,35,39)(27,37,38,30)(28,40,36,34)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=h,e^2=h,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, 8332838);