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 | #49388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,14)(13,18)(15,17)(19,21)(20,23)(22,24)(26,32)(28,38)(33,40)(34,35);(2,5)(6,7)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,21,24,23)(25,26,31,32)(27,33,37,40)(28,29,38,36)(30,34,39,35);(1,2)(3,6)(4,5)(7,8)(10,17)(11,12)(18,24)(19,20)(25,27,31,37)(26,33,32,40)(28,35,38,34)(29,30,36,39);(9,12,15,11)(10,14,17,16)(13,20,22,19)(18,23,24,21)(25,28)(26,29)(27,35)(30,33)(31,38)(32,36)(34,37)(39,40);(1,2)(3,7)(4,5)(6,8)(9,13)(10,18)(11,19)(12,20)(14,23)(15,22)(16,21)(17,24)(25,29)(26,28)(27,30)(31,36)(32,38)(33,35)(34,40)(37,39);(1,3)(2,6)(4,8)(5,7)(9,14)(10,11)(12,17)(13,21)(15,16)(18,20)(19,24)(22,23)(25,30)(26,34)(27,36)(28,33)(29,37)(31,39)(32,35)(38,40)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9686321);