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 | |
4504 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)(3,4,7,6)(10,16)(11,19)(12,21)(18,23)(25,26)(27,36)(28,33)(29,37)(30,38)(31,34)(32,35)(39,40);(1,3,5,7)(2,4,8,6)(9,10,15,16)(11,14,19,24)(12,22,21,17)(13,18,20,23)(25,27,32,34)(26,31,35,36)(28,37,39,30)(29,33,38,40);(1,3)(2,6)(4,8)(5,7)(9,11,15,19)(10,14,16,24)(12,20,21,13)(17,18,22,23)(25,28,32,39)(26,33,35,40)(27,37,34,30)(29,31,38,36);(1,3)(2,4)(5,7)(6,8)(9,12)(10,17)(11,20)(13,19)(14,18)(15,21)(16,22)(23,24)(25,29)(26,30)(27,33)(28,31)(32,38)(34,40)(35,37)(36,39);(9,13,15,20)(10,18,16,23)(11,12,19,21)(14,22,24,17)(25,30)(26,29)(27,28)(31,33)(32,37)(34,39)(35,38)(36,40);(1,4)(2,7)(3,8)(5,6)(9,14)(10,19)(11,16)(12,23)(13,22)(15,24)(17,20)(18,21)(25,31)(26,34)(27,35)(28,38)(29,39)(30,33)(32,36)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8353224);