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 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,15,14)(11,16,22,21)(12,18,13,17)(19,24,20,23)(25,26,30,34)(27,35,38,31)(28,39,37,32)(29,33,36,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,14,21)(12,19,13,20)(17,23,18,24)(25,27,30,38)(26,31,34,35)(28,29,37,36)(32,33,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,14,18)(11,19,22,20)(16,23,21,24)(25,28,30,37)(26,32,34,39)(27,36,38,29)(31,40,35,33);(2,5)(6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(1,2)(3,6)(4,5)(7,8)(9,13,15,12)(10,18,14,17)(11,20,22,19)(16,24,21,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39);(1,3,4,8)(2,6,5,7)(9,14,15,10)(11,21,22,16)(12,18,13,17)(19,24,20,23)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066465);