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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,20)(11,14,19,16)(12,18,13,17)(21,24,22,23)(25,26,31,29)(27,34,30,32)(28,33,38,36)(35,40,37,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,27,31,30)(26,32,29,34)(28,35,38,37)(33,39,36,40);(9,12,15,13)(10,17,20,18)(11,21,19,22)(14,23,16,24)(25,26,31,29)(27,34,30,32)(28,33,38,36)(35,40,37,39);(1,3,5,4)(2,6,8,7)(9,13,15,12)(10,18,20,17)(11,22,19,21)(14,24,16,23)(25,28,31,38)(26,33,29,36)(27,35,30,37)(32,39,34,40);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,21)(13,22)(15,19)(16,20)(17,23)(18,24)(25,29,31,26)(27,34,30,32)(28,36,38,33)(35,40,37,39);(1,4,5,3)(2,7,8,6)(9,14)(10,19)(11,20)(12,23)(13,24)(15,16)(17,22)(18,21)(25,30,31,27)(26,32,29,34)(28,37,38,35)(33,39,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562374);