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 | |
4559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,15)(13,22)(14,18)(16,21)(17,20)(23,24)(25,26,31,35)(27,36,38,32)(28,33,37,29)(30,34,39,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,18,22,23)(25,27,31,38)(26,32,35,36)(28,30,37,39)(29,34,33,40);(1,3)(2,5)(4,8)(6,7)(10,12)(16,19)(17,22)(18,23)(25,28,31,37)(26,33,35,29)(27,30,38,39)(32,40,36,34);(3,8)(5,6)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,18,24,23)(25,28)(26,29)(27,30)(31,37)(32,34)(33,35)(36,40)(38,39);(1,2)(3,5)(4,7)(6,8)(9,13)(10,17)(11,14)(12,22)(15,20)(16,18)(19,23)(21,24)(25,29)(26,28)(27,34)(30,32)(31,33)(35,37)(36,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,22)(17,19)(25,30)(26,34)(27,37)(28,38)(29,36)(31,39)(32,33)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,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, 8609744);