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 | #44882 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,31,32)(27,35,38,30)(28,39,37,33)(29,40,36,34);(1,2,5,6)(3,7,8,4)(10,11)(12,19)(16,18)(20,24)(25,27,31,38)(26,30,32,35)(28,36,37,29)(33,34,39,40);(2,6)(4,7)(13,21)(16,18)(20,24)(22,23)(26,32)(30,35)(33,39)(34,40);(2,6)(3,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28)(26,33)(27,36)(29,38)(30,34)(31,37)(32,39)(35,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,36)(26,34,32,40)(27,37,38,28)(30,33,35,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,35)(26,27,32,38)(28,34,37,40)(29,33,36,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295146);