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 | |
4410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10,14,18)(11,19,22,15)(12,23,21,16)(13,17,20,24)(28,35)(32,37)(34,39)(38,40);(2,4)(7,8)(9,11,14,22)(10,15,18,19)(12,13,21,20)(16,17,23,24)(26,30)(27,33)(32,37)(34,39);(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,24,19,17)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19);(1,4,6,2)(3,8,5,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,24)(21,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,5,6,3)(2,8,4,7)(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,23,19,16)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015980);