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 | |
4618 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)(11,20)(12,14)(13,23)(15,19)(16,24)(17,22)(18,21)(25,26,30,31)(27,28,35,32)(29,33,38,39)(34,36,40,37);(1,2,4,6)(3,5,8,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,18)(19,20)(26,31)(28,32)(33,39)(36,37);(1,2)(3,5)(4,6)(7,8)(25,27)(26,32)(28,31)(29,34)(30,35)(33,37)(36,39)(38,40);(1,3,4,8)(2,5,6,7)(9,12,15,22)(10,17,19,14)(11,21,24,13)(16,18,20,23)(25,28,30,32)(26,27,31,35)(29,37,38,36)(33,40,39,34);(9,13,15,21)(10,18,19,23)(11,22,24,12)(14,16,17,20);(9,14,15,17)(10,12,19,22)(11,23,24,18)(13,16,21,20)(25,29,30,38)(26,33,31,39)(27,34,35,40)(28,36,32,37)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=g,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, 8368377);