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 | |
4402 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)(3,8)(4,5)(6,7)(9,10)(11,16)(12,13)(14,19)(15,17)(18,24)(20,23)(21,22)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,3,6,5)(2,4,7,8)(9,11,15,19)(10,16,17,14)(12,22,18,20)(13,21,24,23)(26,31)(27,34)(33,36)(35,40);(1,3,6,5)(2,4,7,8)(10,17)(11,19)(12,18)(21,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(2,7)(3,5)(9,12,15,18)(10,13,17,24)(11,20,19,22)(14,21,16,23)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,4,6,8)(2,3,7,5)(9,13)(10,18)(11,21)(12,17)(14,22)(15,24)(16,20)(19,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,5,6,3)(2,4,7,8)(9,14,15,16)(10,19,17,11)(12,23,18,21)(13,20,24,22)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627958);