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 | #2408 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4450 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)(9,10,14,18)(11,15,16,12)(13,17,22,21)(19,23,24,20)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3,5,7)(2,4,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(26,30)(31,33)(32,36)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19);(1,4)(2,7)(3,8)(5,6)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,e^2=ghi,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, 8353234);