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 | |
4552 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,36)(27,35,30,32)(28,38,39,33)(29,34,40,37);(1,2)(3,4)(5,7)(6,8)(11,17)(15,21)(18,23)(22,24)(25,27,31,30)(26,32,36,35)(28,37,39,34)(29,38,40,33);(1,3)(2,6)(4,7)(5,8)(10,14)(15,21)(16,19)(22,24)(25,28,31,39)(26,33,36,38)(27,37,30,34)(29,35,40,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,33)(27,37)(29,32)(30,34)(31,39)(35,40)(36,38);(1,4,5,6)(2,3,7,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29)(26,34)(27,33)(28,35)(30,38)(31,40)(32,39)(36,37);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,27)(26,35,36,32)(28,34,39,37)(29,33,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,e^2=ghi,f^2=g,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, 10051783);