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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,8,4,7)(9,10)(11,20)(12,14)(13,23)(15,19)(16,24)(17,22)(18,21)(27,33)(31,34)(35,39)(38,40);(2,6)(7,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,18)(19,20)(25,26,30,29)(27,34,33,31)(28,36,37,32)(35,38,39,40);(1,3,5,4)(2,7,6,8)(25,27,30,33)(26,31,29,34)(28,35,37,39)(32,38,36,40);(1,2)(3,7)(4,8)(5,6)(9,12,15,22)(10,17,19,14)(11,21,24,13)(16,18,20,23)(25,28,30,37)(26,32,29,36)(27,35,33,39)(31,38,34,40);(9,13,15,21)(10,18,19,23)(11,22,24,12)(14,16,17,20);(1,4,5,3)(2,7,6,8)(9,14,15,17)(10,12,19,22)(11,23,24,18)(13,16,21,20)(25,29,30,26)(27,34,33,31)(28,32,37,36)(35,38,39,40)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=ghi,f^2=gi,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, 8370409);