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 | |
4477 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)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26)(27,32)(28,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,2)(3,8)(4,5)(6,7)(9,11,14,13)(10,15,17,18)(12,21,22,19)(16,24,20,23)(25,27,31,30)(26,32,37,36)(28,35,39,38)(29,34,40,33);(1,3,6,5)(2,4,7,8)(25,28)(26,33)(27,38)(29,36)(30,35)(31,39)(32,40)(34,37);(1,4,6,8)(2,3,7,5)(9,12,14,22)(10,16,17,20)(11,19,13,21)(15,23,18,24)(25,28)(26,34)(27,35)(29,36)(30,38)(31,39)(32,40)(33,37);(1,2)(3,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)(25,29)(26,35)(27,34)(28,36)(30,33)(31,40)(32,39)(37,38);(1,5,6,3)(2,4,7,8)(9,13,14,11)(10,15,17,18)(12,19,22,21)(16,24,20,23)(25,30,31,27)(26,36,37,32)(28,35,39,38)(29,34,40,33)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608755);