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 |
| prime radical | #2430 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,15,19)(11,20,21,16)(12,17,22,24)(13,14,18,23)(25,26,31,34)(27,35,37,32)(28,38,36,33)(29,39,40,30);(2,3)(6,7)(11,21)(12,22)(16,20)(17,24)(25,27,31,37)(26,32,34,35)(28,30,36,39)(29,38,40,33);(1,2,5,3)(4,6,8,7)(9,11)(10,16)(12,18)(13,22)(14,17)(15,21)(19,20)(23,24)(25,28,31,36)(26,33,34,38)(27,30,37,39)(29,35,40,32);(1,3,5,2)(4,7,8,6)(9,12,15,22)(10,17,19,24)(11,18,21,13)(14,20,23,16)(25,29,31,40)(26,30,34,39)(27,33,37,38)(28,35,36,32);(1,4,5,8)(2,6,3,7)(9,13,15,18)(10,14,19,23)(11,22,21,12)(16,17,20,24);(9,14)(10,18)(11,17)(12,16)(13,19)(15,23)(20,22)(21,24)(25,30)(26,29)(27,36)(28,37)(31,39)(32,38)(33,35)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=g,d^2=h,e^2=i,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, 8353034);