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 | |
4477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,17)(13,14)(15,19)(16,23)(20,22)(21,24)(25,26,31,29)(27,30,33,39)(28,36,38,32)(34,40,35,37);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,24)(18,19)(25,27,31,33)(26,30,29,39)(28,35,38,34)(32,40,36,37);(1,3)(2,5)(4,8)(6,7)(9,12,15,13)(10,17,19,14)(11,20,23,21)(16,24,18,22)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,35)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,12)(10,17,19,14)(11,21,23,20)(16,24,18,22)(25,29,31,26)(27,30,33,39)(28,36,38,32)(34,37,35,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,18)(12,20)(13,21)(14,24)(15,23)(16,19)(17,22)(25,28)(26,32)(27,35)(29,36)(30,40)(31,38)(33,34)(37,39);(1,2)(3,5)(4,7)(6,8)(9,14,15,17)(10,13,19,12)(11,22,23,24)(16,21,18,20)(25,30)(26,33)(27,29)(28,37)(31,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8565163);