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 | |
4365 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)(3,5)(4,7)(6,8)(9,10,14,15)(11,19,20,16)(12,17,21,24)(13,22,23,18)(25,26)(27,32)(28,34)(29,35)(30,38)(31,33)(36,39)(37,40);(1,2,6,8)(3,5,7,4)(10,15)(12,21)(16,19)(18,22)(25,27,31,36)(26,32,33,39)(28,37,38,29)(30,35,34,40);(1,3,6,7)(2,4,8,5)(10,15)(11,20)(17,24)(18,22)(26,33)(27,36)(28,38)(35,40);(1,4,6,5)(2,3,8,7)(9,11)(10,16)(12,22)(13,17)(14,20)(15,19)(18,21)(23,24)(25,28)(26,34)(27,37)(29,36)(30,33)(31,38)(32,40)(35,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,21)(15,24)(16,23)(20,22)(25,29,31,37)(26,35,33,40)(27,38,36,28)(30,39,34,32);(1,5,6,4)(2,7,8,3)(9,13,14,23)(10,18,15,22)(11,17,20,24)(12,16,21,19)(25,30,31,34)(26,28,33,38)(27,35,36,40)(29,32,37,39)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=ghi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9067945);