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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,22,20,24)(25,26,30,34)(27,35,36,31)(28,32,37,40)(29,33,38,39);(1,2)(3,6)(4,7)(5,8)(10,16)(11,12)(18,21)(19,20)(27,36)(28,37)(31,35)(32,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,22,21,24)(25,27,30,36)(26,31,34,35)(28,38,37,29)(32,33,40,39);(3,8)(5,6)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,24,21,22)(25,28,30,37)(26,32,34,40)(27,29,36,38)(31,39,35,33);(1,2)(3,5)(4,7)(6,8)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,22,17,24)(25,26)(27,31)(28,32)(29,39)(30,34)(33,38)(35,36)(37,40);(1,3)(2,6)(4,8)(5,7)(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,19,21,20)(25,29)(26,33)(27,28)(30,38)(31,40)(32,35)(34,39)(36,37)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8609784);