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 | #59141 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(12,20)(16,23)(19,21)(22,24)(25,26,32,35)(27,36,29,30)(28,37,38,33)(31,39,40,34);(3,8)(6,7)(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,23)(21,22)(25,27,32,29)(26,30,35,36)(28,34,38,39)(31,37,40,33);(1,3)(2,6)(4,7)(5,8)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,23,24)(25,28,32,38)(26,33,35,37)(27,34,29,39)(30,31,36,40);(1,2,5,4)(3,7,8,6)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,32,27)(26,30,35,36)(28,34,38,39)(31,33,40,37);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,30)(26,27)(28,31)(29,35)(32,36)(33,34)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,15,17,18)(12,21,20,19)(16,22,23,24)(25,31)(26,34)(27,37)(28,36)(29,33)(30,38)(32,40)(35,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=h,f^2=i,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, 9620892);