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 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,5)(6,7)(9,10,15,18)(11,13,16,21)(12,14,17,23)(19,20,22,24)(26,31)(28,38)(32,34)(35,36);(1,3,6,5)(2,4,7,8)(9,11,15,16)(10,13,18,21)(12,19,17,22)(14,20,23,24)(25,26)(27,32)(28,33)(29,35)(30,31)(34,37)(36,39)(38,40);(2,7)(4,8)(11,16)(13,21)(19,22)(20,24)(25,27,30,37)(26,32,31,34)(28,36,38,35)(29,33,39,40);(2,7)(3,5)(9,12)(10,14)(11,19)(13,20)(15,17)(16,22)(18,23)(21,24)(25,28)(26,33)(27,36)(29,34)(30,38)(31,40)(32,39)(35,37);(1,4)(2,3)(5,7)(6,8)(9,13,15,21)(10,16,18,11)(12,20,17,24)(14,22,23,19)(25,27)(26,34)(28,36)(29,33)(30,37)(31,32)(35,38)(39,40);(1,5,6,3)(2,4,7,8)(9,14)(10,17)(11,20)(12,18)(13,22)(15,23)(16,24)(19,21)(25,29)(26,35)(27,33)(28,32)(30,39)(31,36)(34,38)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529341);