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 | |
4307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,21,24)(12,13,22,16)(18,23,19,20)(25,26)(27,37)(28,34)(29,30)(31,35)(32,36)(33,39)(38,40);(1,2,5,4)(3,6,8,7)(9,11,14,21)(10,15,17,24)(12,19,22,18)(13,20,16,23)(25,27)(26,33)(28,31)(29,38)(30,35)(32,39)(34,40)(36,37);(1,2,5,4)(3,7,8,6)(9,12,14,22)(10,16,17,13)(11,18,21,19)(15,20,24,23)(25,28,32,29)(26,34,36,30)(27,31,39,38)(33,40,37,35);(1,3,5,8)(2,6,4,7)(9,12)(10,13)(11,19)(14,22)(15,20)(16,17)(18,21)(23,24)(25,29,32,28)(26,34,36,30)(27,38,39,31)(33,40,37,35);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,23)(14,21)(16,20)(17,24)(19,22)(25,30,32,34)(26,29,36,28)(27,35,39,40)(31,33,38,37);(1,4,5,2)(3,7,8,6)(9,13)(10,12)(11,20)(14,16)(15,19)(17,22)(18,24)(21,23)(25,31,32,38)(26,35,36,40)(27,28,39,29)(30,37,34,33)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,d^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, 10066369);