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 | |
4451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,16,20,14)(12,18,22,23)(13,19,21,24)(25,26,32,36)(27,33,38,40)(28,34,29,31)(30,35,37,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,20)(10,16,17,14)(12,21,22,13)(18,24,23,19)(25,27,32,38)(26,33,36,40)(28,39,29,35)(30,31,37,34);(10,17)(11,20)(12,22)(19,24)(25,28,32,29)(26,34,36,31)(27,35,38,39)(30,33,37,40);(2,4)(6,7)(9,12)(10,18)(11,21)(13,20)(14,19)(15,22)(16,24)(17,23)(25,29,32,28)(26,31,36,34)(27,35,38,39)(30,33,37,40);(1,4,5,2)(3,7,8,6)(9,13,15,21)(10,19,17,24)(11,22,20,12)(14,18,16,23)(25,30)(26,35)(27,31)(28,33)(29,40)(32,37)(34,38)(36,39);(1,3)(2,7)(4,6)(5,8)(9,14,15,16)(10,20,17,11)(12,19,22,24)(13,23,21,18)(25,31)(26,28)(27,37)(29,36)(30,38)(32,34)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=hi,d^2=hi,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=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, 10050791);