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 | |
4529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,22)(13,18)(14,16)(15,21)(17,23)(20,24)(25,26)(27,35)(28,33)(29,31)(30,34)(32,36)(37,38)(39,40);(3,7)(6,8)(9,11,14,21)(10,15,16,19)(12,20,23,18)(13,17,24,22)(27,36)(28,37)(32,35)(33,38);(1,2)(3,6)(4,5)(7,8)(10,16)(12,23)(15,19)(18,20)(25,27,31,36)(26,32,29,35)(28,30,37,40)(33,39,38,34);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23)(25,28)(26,33)(27,30)(29,38)(31,37)(32,39)(34,35)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,20)(13,19)(14,23)(15,24)(16,22)(18,21)(25,29,31,26)(27,32,36,35)(28,38,37,33)(30,39,40,34);(9,13,14,24)(10,18,16,20)(11,17,21,22)(12,19,23,15)(25,30)(26,34)(27,37)(28,36)(29,39)(31,40)(32,33)(35,38)Polycyclic presentation:
2^9|b^2=g,c^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440887);