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 | |
4391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17)(11,15,19,14)(12,16,21,20)(18,23,24,22)(25,26,30,29)(27,31,36,34)(28,32,37,38)(33,39,35,40);(1,3,5,7)(2,6,4,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27)(26,31)(28,35)(29,34)(30,36)(32,40)(33,37)(38,39);(2,4)(3,7)(9,10,13,17)(11,14,19,15)(12,20,21,16)(18,23,24,22)(25,28,30,37)(26,32,29,38)(27,33,36,35)(31,39,34,40);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,29,30,26)(27,34,36,31)(28,32,37,38)(33,39,35,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,40)(33,36)(34,39);(1,4,5,2)(3,8,7,6)(9,12,13,21)(10,16,17,20)(11,18,19,24)(14,22,15,23)(25,27)(26,31)(28,35)(29,34)(30,36)(32,40)(33,37)(38,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=i,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, 8900677);