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 | |
4429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,7,8)(9,10)(11,20)(12,14)(13,23)(15,19)(16,24)(17,22)(18,21)(25,26,31,36)(27,37,39,32)(28,29,34,33)(30,35,38,40);(1,3,6,7)(2,4,5,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,18)(19,20)(25,27)(26,32)(28,38)(29,35)(30,34)(31,39)(33,40)(36,37);(1,4)(2,3)(5,7)(6,8)(25,28,31,34)(26,33,36,29)(27,38,39,30)(32,40,37,35);(1,5,6,2)(3,4,7,8)(9,12,15,22)(10,17,19,14)(11,21,24,13)(16,18,20,23)(25,26)(27,37)(28,33)(29,34)(30,40)(31,36)(32,39)(35,38);(9,13,15,21)(10,18,19,23)(11,22,24,12)(14,16,17,20)(25,29,31,33)(26,34,36,28)(27,35,39,40)(30,37,38,32);(1,4)(2,7)(3,5)(6,8)(9,14,15,17)(10,12,19,22)(11,23,24,18)(13,16,21,20)(25,30,31,38)(26,35,36,40)(27,34,39,28)(29,37,33,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=gh,e^2=g,f^2=g,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, 8343746);