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 | |
4371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,5,4,7)(9,10,13,14)(11,15,17,18)(12,20,22,16)(19,24,21,23)(25,26)(27,29)(28,32)(30,31)(33,34)(35,40)(36,39)(37,38);(1,3,6,4)(2,7,8,5)(10,14)(11,17)(16,20)(19,21)(25,26)(27,34)(28,36)(29,33)(30,31)(32,39)(35,40)(37,38);(10,14)(15,18)(16,20)(23,24)(26,31)(27,33)(32,36)(35,37);(2,8)(3,4)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,22)(21,23)(25,27,30,33)(26,29,31,34)(28,37,39,35)(32,38,36,40);(1,4,6,3)(2,7,8,5)(9,11)(10,15)(12,21)(13,17)(14,18)(16,24)(19,22)(20,23)(25,28,30,39)(26,32,31,36)(27,35,33,37)(29,40,34,38);(1,5,6,7)(2,4,8,3)(9,12,13,22)(10,16,14,20)(11,19,17,21)(15,23,18,24)(25,29)(26,33)(27,31)(28,38)(30,34)(32,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154068);