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 | |
4347 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)(3,7,5,8)(25,26)(27,37)(28,33)(29,39)(30,34)(31,35)(32,36)(38,40);(1,3,6,5)(2,7,4,8)(9,10,14,15)(11,16,18,23)(12,13,17,22)(19,24,21,20)(25,27)(26,28)(29,34)(30,31)(32,33)(35,38)(36,37)(39,40);(1,3,6,5)(2,8,4,7)(10,15)(11,18)(13,22)(19,21)(25,28,32,37)(26,33,36,27)(29,40,35,30)(31,34,39,38);(9,11)(10,16)(12,21)(13,20)(14,18)(15,23)(17,19)(22,24)(25,29,32,35)(26,31,36,39)(27,34,33,38)(28,30,37,40);(1,4,6,2)(3,7,5,8)(9,12,14,17)(10,13,15,22)(11,19,18,21)(16,24,23,20)(25,30)(26,34)(27,31)(28,29)(32,40)(33,39)(35,37)(36,38);(1,5,6,3)(2,7,4,8)(9,13)(10,17)(11,20)(12,15)(14,22)(16,19)(18,24)(21,23)(25,31)(26,35)(27,30)(28,38)(29,36)(32,39)(33,40)(34,37)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8826631);