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 | |
4342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(11,17)(14,18)(19,23)(22,24)(25,26)(27,34)(28,29)(30,31)(32,37)(33,38)(35,40)(36,39);(2,5)(3,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,24)(22,23)(26,31)(28,33)(32,34)(35,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,18)(12,15,21,20)(19,22,23,24)(25,26,30,31)(27,32,37,34)(28,29,33,38)(35,36,39,40);(2,5)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,21)(22,23)(25,27,30,37)(26,32,31,34)(28,35,33,39)(29,36,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,17)(15,22)(16,18)(20,24)(21,23)(25,28,30,33)(26,29,31,38)(27,35,37,39)(32,36,34,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,19)(13,21)(14,22)(16,20)(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=ghi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8736361);