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 | #56546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4352 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,13)(14,16)(15,21)(18,23)(19,22)(20,24)(27,30)(28,32)(35,39)(36,40);(1,3,5,4)(2,6,7,8)(9,11,14,21)(10,15,16,17)(12,19,18,24)(13,20,23,22)(25,26)(27,28)(29,37)(30,32)(31,34)(33,38)(35,40)(36,39);(10,16)(12,18)(15,17)(19,24)(25,27,31,30)(26,32,34,28)(29,35,38,39)(33,40,37,36);(9,12,14,18)(10,13,16,23)(11,19,21,24)(15,20,17,22)(25,28,31,32)(26,27,34,30)(29,36,38,40)(33,35,37,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,19)(13,22)(14,21)(15,16)(18,24)(20,23)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,40)(34,37);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,20)(12,16)(14,23)(15,24)(17,19)(21,22)(25,30,31,27)(26,32,34,28)(29,35,38,39)(33,36,37,40)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620562);