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 | #37035 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,14,17,16)(12,21,22,13)(19,20,24,23)(26,31)(29,39)(32,38)(33,40);(1,2)(3,4)(5,7)(6,8)(9,11,15,17)(10,16,18,14)(12,19,22,24)(13,20,21,23)(26,31)(28,37)(32,38)(34,35);(1,3)(2,6)(4,7)(5,8)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4,5,6)(2,3,7,8)(9,12,15,22)(10,13,18,21)(11,19,17,24)(14,23,16,20)(25,27,30,36)(26,32,31,38)(28,34,37,35)(29,40,39,33);(1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,20)(14,24)(15,21)(16,19)(17,23)(18,22)(25,28,30,37)(26,33,31,40)(27,35,36,34)(29,32,39,38);(1,2)(3,6)(4,8)(5,7)(9,14,15,16)(10,17,18,11)(12,20,22,23)(13,19,21,24)(25,29,30,39)(26,34,31,35)(27,33,36,40)(28,38,37,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131396);