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 | #44836 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22)(25,26,31,35)(27,36,38,32)(28,37,39,33)(29,30,34,40);(1,2)(3,5)(4,7)(6,8)(10,14)(15,18)(16,20)(22,24)(25,27,31,38)(26,32,35,36)(28,29,39,34)(30,37,40,33);(1,3)(2,7)(4,6)(5,8)(10,14)(11,17)(12,21)(22,24)(28,39)(29,34)(30,40)(33,37);(2,8)(3,4)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,31,39)(26,33,35,37)(27,34,38,29)(30,32,40,36);(1,4,6,3)(2,5,8,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,29,31,34)(26,30,35,40)(27,28,38,39)(32,33,36,37);(1,5)(2,3)(4,8)(6,7)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,30)(26,34)(27,37)(28,32)(29,35)(31,40)(33,38)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8970464);