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 | |
4725 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)(3,5)(4,7)(6,8)(12,20)(16,21)(18,24)(22,23)(26,31)(27,34)(29,38)(35,37);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26,30,31)(27,32,34,39)(28,29,33,38)(35,40,37,36);(1,2,5,3)(4,6,8,7)(10,14)(15,17)(16,21)(22,23)(27,34)(32,39)(35,37)(36,40);(2,3)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,34)(26,32,31,39)(28,37,33,35)(29,36,38,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,33)(26,29,31,38)(27,35,34,37)(32,40,39,36);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,36)(28,31)(30,38)(32,35)(34,40)(37,39)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8062219);