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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,17,20,16)(12,13,21,22)(18,23,24,19)(25,26,31,34)(27,29,37,32)(28,30,38,33)(35,40,39,36);(2,5)(6,8)(10,15)(12,21)(16,17)(18,24)(25,27,31,37)(26,32,34,29)(28,35,38,39)(30,36,33,40);(1,2)(3,6)(4,5)(7,8)(25,28,31,38)(26,33,34,30)(27,35,37,39)(29,36,32,40);(9,11,14,20)(10,16,15,17)(12,18,21,24)(13,19,22,23)(25,27)(26,32)(28,35)(29,34)(30,36)(31,37)(33,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,18)(14,21)(15,22)(16,19)(17,23)(20,24)(25,29,31,32)(26,27,34,37)(28,36,38,40)(30,39,33,35);(1,3)(2,6)(4,7)(5,8)(9,13)(10,12)(11,19)(14,22)(15,21)(16,18)(17,24)(20,23)(25,30)(26,28)(27,36)(29,39)(31,33)(32,35)(34,38)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015563);