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 | #2410 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4404 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)(11,19)(12,21)(13,20)(14,18)(15,16)(17,23)(22,24)(25,26,31,35)(27,29,36,32)(28,33,37,38)(30,34,39,40);(2,5)(6,8)(9,11,15,20)(10,13,16,19)(12,22,23,14)(17,24,21,18)(27,36)(28,37)(29,32)(33,38);(1,2,4,5)(3,6,7,8)(10,16)(13,19)(17,21)(18,24)(25,27,31,36)(26,32,35,29)(28,30,37,39)(33,40,38,34);(3,7)(6,8)(9,12,15,23)(10,17,16,21)(11,14,20,22)(13,18,19,24)(25,28)(26,33)(27,30)(29,34)(31,37)(32,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,18)(14,17)(15,19)(16,20)(21,22)(23,24)(25,29,31,32)(26,27,35,36)(28,34,37,40)(30,38,39,33);(1,3)(2,6)(4,7)(5,8)(9,14)(10,18)(11,12)(13,17)(15,22)(16,24)(19,21)(20,23)(25,30)(26,34)(27,37)(28,36)(29,38)(31,39)(32,33)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506698);