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 | |
4779 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,21)(22,24)(25,26)(27,34)(28,32)(29,37)(30,31)(33,38)(35,39)(36,40);(1,3,5,4)(2,6,7,8)(9,11,15,14)(10,16,12,18)(13,19,21,22)(17,23,20,24)(26,31)(32,34)(33,37)(39,40);(9,10,15,12)(11,18,14,16)(13,17,21,20)(19,24,22,23)(25,27,30,28)(26,32,31,34)(29,35,38,36)(33,39,37,40);(1,2)(3,8)(4,6)(5,7)(9,12,15,10)(11,16,14,18)(13,17,21,20)(19,24,22,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,28,30,27)(26,32,31,34)(29,35,38,36)(33,40,37,39);(1,2)(3,6)(4,8)(5,7)(9,14,15,11)(10,16,12,18)(13,19,21,22)(17,24,20,23)(25,29,30,38)(26,33,31,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10389700);