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 | |
4672 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)(26,31)(27,33)(32,39)(34,36);(1,2,5,4)(3,7,8,6)(10,15)(12,23)(16,19)(18,21)(25,26,30,31)(27,29,33,38)(28,32,35,39)(34,37,36,40);(1,3,5,8)(2,6,4,7)(10,15)(11,20)(12,23)(13,24)(27,33)(28,35)(29,38)(32,39);(3,8)(6,7)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,24)(21,23)(25,27,30,33)(26,29,31,38)(28,36,35,34)(32,40,39,37);(1,3)(2,7)(4,6)(5,8)(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40);(1,4,5,2)(3,6,8,7)(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,29)(26,33)(27,31)(28,37)(30,38)(32,36)(34,39)(35,40)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,d^2=gh,f^2=i,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, 8935334);