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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,22)(12,15)(13,20)(14,19)(16,21)(17,24)(18,23)(25,26)(27,31)(28,30)(29,32)(33,35)(34,39)(36,38)(37,40);(1,2,5,4)(3,6,8,7)(9,11)(10,17)(12,13)(14,23)(15,19)(16,24)(18,20)(21,22)(26,28)(31,33)(32,36)(39,40);(2,4)(6,7)(9,12,16,23)(10,18,21,15)(11,13,24,14)(17,20,22,19)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(9,13,16,14)(10,19,21,20)(11,12,24,23)(15,22,18,17)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,3,5,8)(2,6,4,7)(9,14,16,13)(10,20,21,19)(11,23,24,12)(15,17,18,22);(1,4,5,2)(3,7,8,6)(9,15,16,18)(10,12,21,23)(11,19,24,20)(13,22,14,17)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103164);