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 |
| prime radical | #50092 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4231 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)(9,10,15,12)(11,16,18,14)(13,17,21,20)(19,23,24,22)(27,30)(32,34)(35,37)(39,40);(1,2,6,8)(3,5,7,4)(9,11,15,18)(10,16,12,14)(13,19,21,24)(17,23,20,22)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,3,6,7)(2,4,8,5)(11,18)(13,21)(14,16)(17,20)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,4,6,5)(2,3,8,7)(9,12,15,10)(11,14,18,16)(13,17,21,20)(19,23,24,22)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(1,2)(3,4)(5,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(1,5,6,4)(2,7,8,3)(9,14)(10,11)(12,18)(13,22)(15,16)(17,19)(20,24)(21,23)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8864753);