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 | |
4478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,30,32)(27,35,37,31)(28,33,39,36)(29,38,40,34);(1,2,5,3)(4,6,8,7)(10,11)(15,17)(16,18)(22,23)(25,27,30,37)(26,31,32,35)(28,38,39,34)(29,36,40,33);(2,3)(6,7)(12,19)(15,17)(20,24)(22,23)(26,32)(28,39)(31,35)(34,38);(2,3)(4,8)(9,11,14,10)(12,15,19,17)(13,16,21,18)(20,23,24,22)(25,27)(26,31)(28,38)(29,33)(30,37)(32,35)(34,39)(36,40);(1,3,5,2)(4,7,8,6)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,33)(27,34)(29,35)(30,39)(31,40)(32,36)(37,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,40)(26,34,32,38)(27,36,37,33)(28,35,39,31)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9012477);