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 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(9,10,15,17)(11,20,21,16)(12,22,19,14)(13,18,23,24)(25,26,32,36)(27,33,38,40)(28,34,29,31)(30,35,37,39);(3,4)(7,8)(9,11)(10,16)(12,13)(14,18)(15,21)(17,20)(19,23)(22,24)(25,27,32,38)(26,33,36,40)(28,39,29,35)(30,31,37,34);(1,3,6,4)(2,7,5,8)(10,17)(11,21)(12,19)(18,24)(25,28,32,29)(26,34,36,31)(27,35,38,39)(30,33,37,40);(3,4)(7,8)(9,12,15,19)(10,14,17,22)(11,13,21,23)(16,18,20,24)(25,29,32,28)(26,31,36,34)(27,35,38,39)(30,33,37,40);(1,4,6,3)(2,7,5,8)(9,13)(10,18)(11,19)(12,21)(14,20)(15,23)(16,22)(17,24)(25,30)(26,35)(27,31)(28,33)(29,40)(32,37)(34,38)(36,39);(1,5,6,2)(3,8,4,7)(9,14)(10,19)(11,18)(12,17)(13,20)(15,22)(16,23)(21,24)(25,31)(26,28)(27,37)(29,36)(30,38)(32,34)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050787);