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 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,12)(11,16,19,20)(13,17,23,22)(14,21,24,18);(1,3,7,6)(2,4,8,5)(9,11)(10,16)(12,20)(13,18)(14,17)(15,19)(21,23)(22,24)(25,26,30,34)(27,32,37,31)(28,29,33,39)(35,40,38,36);(1,4,7,5)(2,3,8,6)(10,12)(11,19)(17,22)(18,21)(25,27,30,37)(26,31,34,32)(28,35,33,38)(29,36,39,40);(2,8)(4,5)(9,12,15,10)(11,20,19,16)(13,17,23,22)(14,21,24,18)(25,27)(26,32)(28,38)(29,36)(30,37)(31,34)(33,35)(39,40);(1,5,7,4)(2,3,8,6)(9,13)(10,17)(11,21)(12,22)(14,20)(15,23)(16,24)(18,19)(25,28)(26,29)(27,35)(30,33)(31,36)(32,40)(34,39)(37,38);(1,6,7,3)(2,5,8,4)(9,14)(10,18)(11,17)(12,21)(13,16)(15,24)(19,22)(20,23)(25,29)(26,33)(27,36)(28,34)(30,39)(31,38)(32,35)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9050585);