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 | |
4348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,14,16)(11,15,17,12)(13,21,23,18)(19,20,22,24)(26,30)(28,35)(31,36)(34,39);(2,5)(3,8)(9,11)(10,15)(12,16)(13,22)(14,17)(18,20)(19,23)(21,24)(27,33)(31,36)(34,39)(38,40);(1,3,6,8)(2,7,5,4)(10,16)(11,17)(18,21)(19,22)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,16)(13,19)(14,17)(18,24)(20,21)(22,23);(1,5,6,2)(3,4,8,7)(9,12,14,15)(10,17,16,11)(13,20,23,24)(18,22,21,19)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4)(2,3)(5,8)(6,7)(9,13,14,23)(10,18,16,21)(11,19,17,22)(12,20,15,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8796569);