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 | |
4457 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,22)(13,21)(14,16)(15,17)(18,23)(19,24)(26,31)(27,34)(28,36)(37,39);(2,5)(6,7)(9,11,15,14)(10,16,17,20)(12,19,23,21)(13,18,24,22)(25,26,30,31)(27,29,34,32)(28,33,36,38)(35,37,40,39);(1,2)(3,6)(4,5)(7,8)(10,17)(12,23)(16,20)(19,21)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,3)(2,6)(4,8)(5,7)(9,12,15,23)(10,18,17,22)(11,21,14,19)(13,20,24,16)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,13)(10,19)(11,18)(12,20)(14,22)(15,24)(16,23)(17,21)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(9,14,15,11)(10,16,17,20)(12,21,23,19)(13,18,24,22)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=gi,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, 8626160);