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 | |
4696 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)(25,26,31,33)(27,30,28,32)(29,34,38,40)(35,37,36,39);(1,3,6,4)(2,7,8,5)(10,14)(11,17)(16,20)(19,21)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,37,40,39);(10,14)(15,18)(16,20)(23,24)(26,33)(27,28)(34,40)(35,36);(2,8)(3,4)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,22)(21,23)(25,28,31,27)(26,30,33,32)(29,36,38,35)(34,37,40,39);(1,4,6,3)(2,7,8,5)(9,11,13,17)(10,15,14,18)(12,21,22,19)(16,24,20,23)(25,29,31,38)(26,34,33,40)(27,35,28,36)(30,37,32,39);(1,5,6,7)(2,4,8,3)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,21)(18,24)(25,30)(26,27)(28,33)(29,37)(31,32)(34,35)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=i,e^2=g,f^2=gh,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, 10154650);