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 | |
4482 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22)(25,26)(27,35)(28,39)(29,37)(30,36)(31,34)(32,38)(33,40);(2,6)(3,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,26,31,34)(27,32,38,35)(28,39,36,30)(29,33,40,37);(1,3)(2,4)(5,8)(6,7)(25,27,31,38)(26,32,34,35)(28,29,36,40)(30,37,39,33);(1,3)(2,4)(5,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,28)(26,30)(27,29)(31,36)(32,37)(33,35)(34,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,29)(26,33)(27,36)(28,38)(30,32)(31,40)(34,37)(35,39);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,30)(26,28)(27,37)(29,32)(31,39)(33,38)(34,36)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8046674);