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 | |
4357 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)(3,7,5,4)(9,10)(11,16)(12,15)(13,17)(14,22)(18,24)(19,20)(21,23)(28,36)(31,38)(35,39)(37,40);(1,3,6,5)(2,4,8,7)(9,11)(10,16)(12,20)(13,22)(14,17)(15,19)(18,23)(21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(1,4)(2,5)(3,8)(6,7)(10,12)(11,19)(17,21)(18,22)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,37,38,40);(1,3,6,5)(2,7,8,4)(9,12,15,10)(11,20,19,16)(13,21,23,17)(14,22,24,18)(25,28)(26,31)(27,35)(29,37)(30,36)(32,39)(33,38)(34,40);(1,4)(2,3)(5,8)(6,7)(9,13,15,23)(10,17,12,21)(11,18,19,22)(14,16,24,20)(25,26)(27,29)(28,31)(30,33)(32,34)(35,37)(36,38)(39,40);(1,5,6,3)(2,7,8,4)(9,14)(10,18)(11,17)(12,22)(13,20)(15,24)(16,23)(19,21)(25,29)(26,32)(27,33)(28,37)(30,34)(31,39)(35,38)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620955);