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 | |
4303 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,5,7,4)(9,10,15,17)(11,20,14,16)(12,18,22,24)(13,19,23,21)(27,33)(31,37)(34,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,17,20)(12,21,22,19)(13,18,23,24)(26,30)(27,33)(32,35)(34,39);(1,4,6,5)(2,7,8,3)(10,17)(12,22)(16,20)(19,21)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,18)(11,19)(13,16)(14,21)(15,22)(17,24)(20,23)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,5,6,4)(2,7,8,3)(9,13)(10,19)(11,18)(12,16)(14,24)(15,23)(17,21)(20,22)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,2)(3,5)(4,7)(6,8)(9,14,15,11)(10,20,17,16)(12,21,22,19)(13,18,23,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9852767);