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 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,7)(9,10)(11,14)(12,17)(13,18)(15,16)(19,24)(20,22)(21,23)(25,26,31,34)(27,32,30,40)(28,33,38,29)(35,36,37,39);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,20)(13,21)(15,19)(16,24)(17,22)(18,23)(25,27,31,30)(26,32,34,40)(28,35,38,37)(29,39,33,36);(1,2)(3,8)(4,7)(5,6)(10,16)(11,19)(12,21)(18,22)(25,28,31,38)(26,29,34,33)(27,35,30,37)(32,39,40,36);(1,3,5,7)(2,4,6,8)(9,12,15,21)(10,17,16,23)(11,20,19,13)(14,22,24,18)(25,28)(26,33)(27,35)(29,34)(30,37)(31,38)(32,36)(39,40);(1,4,5,8)(2,7,6,3)(9,13,15,20)(10,18,16,22)(11,12,19,21)(14,17,24,23)(25,29,31,33)(26,28,34,38)(27,36,30,39)(32,37,40,35);(1,2)(3,4)(5,6)(7,8)(9,14,15,24)(10,19,16,11)(12,22,21,18)(13,23,20,17)(25,30,31,27)(26,32,34,40)(28,37,38,35)(29,39,33,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050578);