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 | |
4499 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)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,23)(21,22);(1,3,6,5)(2,7,4,8)(9,11,15,12)(10,16,17,18)(13,20,24,22)(14,21,19,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,6,5)(2,8,4,7)(10,17)(14,19)(16,18)(21,23)(26,30)(31,33)(32,37)(38,39);(1,4,6,2)(3,7,5,8)(9,12,15,11)(10,18,17,16)(13,22,24,20)(14,23,19,21)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,5,6,3)(2,7,4,8)(9,13)(10,19)(11,20)(12,22)(14,17)(15,24)(16,23)(18,21)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,14,15,19)(10,13,17,24)(11,21,12,23)(16,20,18,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=hi,e^2=ghi,f^2=g,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, 8101050);