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 | C2×#120 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,31,35)(27,33,28,32)(29,30,34,40)(36,39,38,37);(1,2,5,8)(3,4,7,6)(10,15)(12,18)(17,21)(20,22)(25,27,31,28)(26,32,35,33)(29,36,34,38)(30,37,40,39);(1,3)(2,6)(4,8)(5,7)(10,15)(11,16)(17,21)(19,24);(9,11,14,16)(10,12,15,18)(13,19,23,24)(17,20,21,22)(25,28,31,27)(26,33,35,32)(29,38,34,36)(30,39,40,37);(1,3)(2,4)(5,7)(6,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,29)(26,30)(27,36)(28,38)(31,34)(32,37)(33,39)(35,40);(1,4)(2,7)(3,8)(5,6)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,38)(33,36)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9583519);