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 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,15,17)(11,19,22,16)(12,13,23,24)(14,18,21,20)(25,26,31,35)(27,32,37,39)(28,29,38,33)(30,34,36,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,22)(10,16,17,19)(12,20,23,18)(13,21,24,14)(25,27,31,37)(26,32,35,39)(28,30,38,36)(29,34,33,40);(10,17)(12,23)(16,19)(18,20)(25,28,31,38)(26,33,35,29)(27,36,37,30)(32,34,39,40);(9,12)(10,13)(11,20)(14,19)(15,23)(16,21)(17,24)(18,22)(25,28)(26,33)(27,30)(29,35)(31,38)(32,40)(34,39)(36,37);(1,3,5,4)(2,6,8,7)(9,13)(10,12)(11,21)(14,22)(15,24)(16,20)(17,23)(18,19)(25,29,31,33)(26,28,35,38)(27,34,37,40)(30,39,36,32);(1,4,5,3)(2,7,8,6)(9,14,15,21)(10,18,17,20)(11,13,22,24)(12,19,23,16)(25,30)(26,34)(27,28)(29,32)(31,36)(33,39)(35,40)(37,38)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7881669);