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 | #51565 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,26)(27,35)(28,30)(29,32)(31,33)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(10,11)(15,17)(16,18)(22,23)(25,27,31,29)(26,32,33,35)(28,36,39,38)(30,37,34,40);(2,5)(6,7)(13,21)(16,18)(19,24)(22,23)(26,33)(30,34)(32,35)(37,40);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28)(26,34)(27,36)(29,38)(30,33)(31,39)(32,40)(35,37);(1,3)(2,6)(4,8)(5,7)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,27)(26,35,33,32)(28,38,39,36)(30,40,34,37);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30,31,34)(26,28,33,39)(27,37,29,40)(32,36,35,38)Polycyclic presentation:
2^9|b^2=hi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095316);