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 | |
4597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(10,14)(11,17)(16,21)(18,23)(25,26,32,34)(27,35,31,30)(28,33,29,39)(36,40,38,37);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,32,31)(26,30,34,35)(28,36,29,38)(33,37,39,40);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,32,29)(26,33,34,39)(27,36,31,38)(30,37,35,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,29,32,28)(26,33,34,39)(27,36,31,38)(30,40,35,37);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,30,32,35)(26,27,34,31)(28,37,29,40)(33,36,39,38);(1,4,5,3)(2,6,7,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,31,32,27)(26,30,34,35)(28,38,29,36)(33,37,39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8603479);