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 |
| prime radical | #57302 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(25,26)(27,34)(28,38)(29,39)(30,37)(31,35)(32,40)(33,36);(3,7)(6,8)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,27,31,37)(26,30,35,34)(28,36,40,29)(32,33,38,39);(1,2)(3,6)(4,5)(7,8)(10,15)(13,22)(16,23)(20,24)(25,28,31,40)(26,32,35,38)(27,36,37,29)(30,33,34,39);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,29)(26,33)(27,28)(30,38)(31,36)(32,34)(35,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,27)(26,34)(28,36)(29,40)(30,35)(31,37)(32,39)(33,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,30,31,34)(26,27,35,37)(28,39,40,33)(29,38,36,32)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8809666);