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 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,15,14)(11,18,17,13)(12,16,20,22)(19,24,23,21)(26,30)(27,33)(32,35)(34,39);(1,2,6,5)(3,8,4,7)(12,20)(16,22)(19,23)(21,24)(26,30)(31,37)(32,35)(38,40);(1,3,6,4)(2,7,5,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,23)(22,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,4,6,3)(2,8,5,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24);(1,5,6,2)(3,8,4,7)(9,13,15,18)(10,17,14,11)(12,21,20,24)(16,23,22,19)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(9,14,15,10)(11,18,17,13)(12,22,20,16)(19,24,23,21)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=h,f^2=gh,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, 8087597);