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 | |
4243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,17)(16,19)(18,23)(20,24)(21,22)(25,26,31,30)(27,34,37,32)(28,29,39,33)(35,36,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,20)(13,19)(15,24)(16,17)(18,22)(21,23)(25,27,31,37)(26,32,30,34)(28,38,39,35)(29,36,33,40);(2,5)(6,8)(9,10,13,17)(11,16,19,14)(12,15,22,21)(18,24,20,23)(25,28,31,39)(26,33,30,29)(27,35,37,38)(32,36,34,40);(2,5)(3,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,24)(17,21)(19,20)(25,29)(26,28)(27,36)(30,39)(31,33)(32,38)(34,35)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,21)(13,17)(14,19)(15,22)(18,23)(20,24)(25,30,31,26)(27,32,37,34)(28,29,39,33)(35,36,38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,20)(13,19)(14,17)(15,23)(18,22)(21,24)(25,29)(26,28)(27,36)(30,39)(31,33)(32,38)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8985989);