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 | |
4525 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(3,7)(6,8)(10,11)(15,17)(16,18)(23,24)(26,32)(27,28)(34,37)(35,36);(1,2)(3,6)(4,5)(7,8)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,3)(2,6)(4,7)(5,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=g,e^2=g,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, 8602566);