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 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,8)(4,6)(9,10,15,19)(11,16,20,22)(12,17,21,24)(13,14,18,23)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(1,2)(3,4)(5,7)(6,8)(9,11,15,20)(10,16,19,22)(12,14,21,23)(13,17,18,24)(26,32)(27,28)(34,37)(35,36);(1,3,5,8)(2,6,7,4)(11,20)(12,21)(16,22)(17,24)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(2,7)(3,8)(9,12)(10,17)(11,14)(13,22)(15,21)(16,18)(19,24)(20,23)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(1,4)(2,3)(5,6)(7,8)(9,13)(10,14)(11,17)(12,22)(15,18)(16,21)(19,23)(20,24)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(1,2)(3,6)(4,8)(5,7)(9,14)(10,18)(11,21)(12,20)(13,19)(15,23)(16,24)(17,22)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=h,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=dh,d^c=dgi,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, 8619382);