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 | |
4189 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)(4,7)(6,8)(9,10)(11,16)(12,14)(13,22)(15,17)(18,21)(19,24)(20,23)(26,30)(28,35)(31,36)(34,39);(1,2,6,8)(3,5,7,4)(9,11)(10,16)(12,21)(13,19)(14,18)(15,20)(17,23)(22,24)(27,33)(31,36)(34,39)(38,40);(1,3,6,7)(2,4,8,5)(10,17)(11,20)(12,19)(18,22)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4,6,5)(2,3,8,7)(9,12,15,19)(10,14,17,24)(11,21,20,13)(16,18,23,22);(1,2)(3,4)(5,7)(6,8)(9,13,15,21)(10,18,17,22)(11,19,20,12)(14,23,24,16)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,5,6,4)(2,7,8,3)(9,14)(10,19)(11,22)(12,17)(13,23)(15,24)(16,21)(18,20)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662244);