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 | |
4383 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,21)(13,17)(15,19)(16,18)(20,24)(22,23)(25,26)(27,33)(28,34)(29,40)(30,35)(31,36)(32,37)(38,39);(1,3,5,6)(2,4,7,8)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,24)(21,22)(25,27,32,30)(26,33,37,35)(28,36,38,29)(31,39,40,34);(9,12,15,13)(10,17,19,21)(11,20,18,23)(14,22,16,24)(25,28)(26,34)(27,29)(30,36)(31,35)(32,38)(33,40)(37,39);(1,2)(3,8)(4,6)(5,7)(9,13,15,12)(10,17,19,21)(11,20,18,23)(14,24,16,22)(25,29,32,36)(26,31,37,40)(27,38,30,28)(33,34,35,39);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,20)(13,23)(15,18)(16,19)(17,22)(21,24)(25,30,32,27)(26,35,37,33)(28,36,38,29)(31,39,40,34);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,19)(12,22)(13,24)(15,16)(17,23)(20,21)(25,31)(26,36)(27,39)(28,35)(29,37)(30,34)(32,40)(33,38)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10428511);