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 | |
4293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,16)(11,19,13,17)(12,14,18,24)(20,23,22,21)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,2)(3,6)(4,7)(5,8)(10,16)(12,18)(17,19)(20,22)(26,31)(27,34)(33,36)(35,40);(9,11,15,13)(10,17,16,19)(12,20,18,22)(14,21,24,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,3,5,4)(2,6,8,7)(9,12,15,18)(10,14,16,24)(11,20,13,22)(17,21,19,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,4,5,3)(2,7,8,6)(9,13,15,11)(10,17,16,19)(12,22,18,20)(14,21,24,23);(9,14)(10,18)(11,21)(12,16)(13,23)(15,24)(17,22)(19,20)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565646);