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 | |
4406 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)(6,8)(9,10)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,40)(36,39);(3,7)(6,8)(10,12)(13,21)(15,17)(19,24)(26,31)(28,33)(32,34)(35,39);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,24)(22,23)(25,26,30,31)(27,32,37,34)(28,29,33,38)(35,36,39,40);(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27,30,37)(26,32,31,34)(28,35,33,39)(29,36,38,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,28,30,33)(26,29,31,38)(27,35,37,39)(32,36,34,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,36)(28,31)(30,38)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662027);