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 | |
4763 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)(11,17)(15,18)(19,23)(22,24)(26,30)(27,33)(32,35)(34,39);(1,2,6,8)(3,5,7,4)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(1,3,6,7)(2,4,8,5)(10,14)(15,18)(16,20)(22,24)(27,33)(31,37)(34,39)(38,40);(1,4,6,5)(2,3,8,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,5,6,4)(2,7,8,3)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659055);