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 | |
4533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,8,7)(9,10)(11,16)(12,13)(14,19)(15,17)(18,24)(20,23)(21,22)(25,26,31,32)(27,29,28,33)(30,34,38,39)(35,37,36,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,19)(10,16,17,14)(12,22,18,20)(13,21,24,23)(26,32)(27,28)(34,39)(35,36);(1,2)(3,7)(4,8)(5,6)(10,17)(11,19)(12,18)(21,23)(25,27,31,28)(26,33,32,29)(30,35,38,36)(34,40,39,37);(2,6)(4,7)(9,12,15,18)(10,13,17,24)(11,20,19,22)(14,21,16,23)(25,28,31,27)(26,29,32,33)(30,36,38,35)(34,37,39,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,18)(11,21)(12,17)(14,22)(15,24)(16,20)(19,23)(25,29)(26,27)(28,32)(30,37)(31,33)(34,35)(36,39)(38,40);(1,4,5,7)(2,8,6,3)(9,14,15,16)(10,19,17,11)(12,23,18,21)(13,20,24,22)(25,30,31,38)(26,34,32,39)(27,35,28,36)(29,37,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=i,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, 8626155);