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 | #36791 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10,15,19)(11,20,21,16)(12,17,22,24)(13,14,18,23)(25,26,30,31)(27,33,36,32)(28,37,38,29)(34,35,40,39);(1,2,6,5)(3,8,4,7)(11,21)(12,22)(16,20)(17,24)(26,31)(28,38)(32,33)(34,40);(1,3,6,4)(2,7,5,8)(9,11)(10,16)(12,18)(13,22)(14,17)(15,21)(19,20)(23,24)(25,27,30,36)(26,32,31,33)(28,34,38,40)(29,35,37,39);(1,4,6,3)(2,8,5,7)(9,12,15,22)(10,17,19,24)(11,18,21,13)(14,20,23,16)(25,28,30,38)(26,29,31,37)(27,34,36,40)(32,35,33,39);(1,5,6,2)(3,8,4,7)(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)(25,29)(26,28)(27,35)(30,37)(31,38)(32,34)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,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, 8353036);