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 | |
4573 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)(27,35)(28,38)(29,34)(30,32)(31,33)(36,40)(37,39);(1,2)(3,4)(5,7)(6,8)(11,17)(15,21)(18,23)(22,24)(25,27,31,30)(26,32,33,35)(28,36,34,39)(29,40,38,37);(1,3)(2,6)(4,7)(5,8)(10,14)(15,21)(16,19)(22,24)(26,33)(28,34)(32,35)(36,39);(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)(26,29)(27,36)(30,39)(31,34)(32,40)(33,38)(35,37);(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)(27,37)(28,33)(30,40)(31,38)(32,36)(35,39);(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)(26,35,33,32)(28,39,34,36)(29,37,38,40)Polycyclic presentation:
2^9|b^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, 10051778);