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 | |
4637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,23,22)(13,21,24,18)(25,26,30,32)(27,31,35,40)(28,33,37,36)(29,38,39,34);(1,2)(3,8)(4,6)(5,7)(10,15)(12,23)(16,19)(18,21)(25,27,30,35)(26,31,32,40)(28,34,37,38)(29,36,39,33);(1,3)(2,6)(4,5)(7,8)(10,15)(11,20)(12,23)(13,24)(26,32)(27,35)(28,37)(29,39);(3,4)(6,8)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,24)(21,23)(25,28)(26,33)(27,34)(29,31)(30,37)(32,36)(35,38)(39,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,27)(26,31)(28,38)(29,36)(30,35)(32,40)(33,39)(34,37);(1,2)(3,8)(4,6)(5,7)(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,29,30,39)(26,34,32,38)(27,36,35,33)(28,31,37,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8906602);