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 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,15)(11,20,22,17)(12,18,13,23)(14,19,21,24)(25,26,31,29)(27,30,33,39)(28,36,38,32)(34,40,35,37);(1,2)(3,6)(4,7)(5,8)(9,11,16,22)(10,17,15,20)(12,21,13,14)(18,19,23,24)(25,27,31,33)(26,30,29,39)(28,35,38,34)(32,40,36,37);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,18,15,23)(11,21,22,14)(17,19,20,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,35)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,18,15,23)(11,14,22,21)(17,19,20,24)(25,29,31,26)(27,30,33,39)(28,36,38,32)(34,37,35,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,19)(11,12)(13,22)(15,24)(16,21)(17,23)(18,20)(25,28)(26,32)(27,35)(29,36)(30,40)(31,38)(33,34)(37,39);(1,2)(3,5)(4,7)(6,8)(9,15,16,10)(11,20,22,17)(12,23,13,18)(14,19,21,24)(25,30)(26,33)(27,29)(28,37)(31,39)(32,35)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,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, 8565166);