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 | |
4578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(3,5)(12,20)(16,23)(18,24)(21,22)(25,26)(27,32)(28,38)(29,33)(30,39)(31,34)(35,36)(37,40);(1,2,6,4)(3,7,5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(25,27,31,37)(26,32,34,40)(28,29,39,35)(30,36,38,33);(1,3,6,5)(2,7,4,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,28)(26,30)(27,35)(29,37)(31,39)(32,33)(34,38)(36,40);(1,3,6,5)(2,8,4,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29,31,35)(26,33,34,36)(27,28,37,39)(30,32,38,40);(1,4,6,2)(3,8,5,7)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,28)(26,30)(27,29)(31,39)(32,36)(33,40)(34,38)(35,37);(1,5,6,3)(2,8,4,7)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,28)(27,36)(29,32)(31,38)(33,37)(34,39)(35,40)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=h,e^2=g,f^2=g,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, 8042849);