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 | |
4690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,15,17)(11,18,12,16)(13,14,23,24)(19,22,21,20)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(1,3,6,5)(2,7,4,8)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(9,11,15,12)(10,16,17,18)(13,19,23,21)(14,20,24,22)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(1,2,6,4)(3,8,5,7)(9,12,15,11)(10,16,17,18)(13,19,23,21)(14,22,24,20)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,4,6,2)(3,8,5,7)(9,13)(10,14)(11,19)(12,21)(15,23)(16,20)(17,24)(18,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,5,6,3)(2,7,4,8)(9,14)(10,13)(11,20)(12,22)(15,24)(16,19)(17,23)(18,21)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=h,d^2=ghi,e^2=gi,f^2=g,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, 10428508);