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 | |
4433 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)(3,7)(4,6)(5,8)(9,10,16,14)(11,21,22,17)(12,19,13,18)(15,20,24,23)(25,26,31,30)(27,32,37,35)(28,33,38,40)(29,39,36,34);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,20)(13,23)(14,21)(15,19)(16,22)(18,24)(25,26,31,30)(27,35,37,32)(28,33,38,40)(29,34,36,39);(9,12,16,13)(10,18,14,19)(11,20,22,23)(15,17,24,21)(25,27,31,37)(26,32,30,35)(28,36,38,29)(33,34,40,39);(2,6)(5,7)(9,13,16,12)(10,19,14,18)(11,20,22,23)(15,17,24,21)(25,28)(26,33)(27,36)(29,37)(30,40)(31,38)(32,34)(35,39);(1,3)(2,5)(4,8)(6,7)(9,14,16,10)(11,21,22,17)(12,19,13,18)(15,23,24,20)(25,29,31,36)(26,34,30,39)(27,28,37,38)(32,40,35,33);(1,2)(3,5)(4,6)(7,8)(9,15)(10,20)(11,19)(12,17)(13,21)(14,23)(16,24)(18,22)(25,30,31,26)(27,35,37,32)(28,33,38,40)(29,39,36,34)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8374401);