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 | |
4466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,8,7)(9,10,16,20)(11,19,15,17)(12,18,13,14)(21,23,22,24)(28,35)(32,37)(34,39)(38,40);(1,2,4,6)(3,7,8,5)(9,11,16,15)(10,17,20,19)(12,22,13,21)(14,24,18,23)(26,30)(27,33)(32,37)(34,39);(1,2)(3,5)(4,6)(7,8)(9,12,16,13)(10,14,20,18)(11,21,15,22)(17,23,19,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,4,8)(2,5,6,7)(9,13,16,12)(10,18,20,14)(11,22,15,21)(17,24,19,23);(9,14,16,18)(10,12,20,13)(11,23,15,24)(17,21,19,22)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,15,16,11)(10,19,20,17)(12,22,13,21)(14,24,18,23)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000566);