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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4404 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)(11,15,18,22)(12,16,21,19)(17,23,20,24)(25,26,29,32)(27,33,35,30)(28,31,38,37)(34,40,36,39);(1,2)(3,7)(4,8)(5,6)(10,14)(15,22)(16,19)(23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,40)(37,38);(1,3)(2,4)(5,8)(6,7)(25,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,40)(37,38);(2,6)(4,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,27,29,35)(26,30,32,33)(28,36,38,34)(31,40,37,39);(1,3)(2,7)(4,6)(5,8)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,26)(27,30)(28,37)(29,32)(31,38)(33,35)(34,40)(36,39);(1,4,5,7)(2,8,6,3)(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24)(25,28,29,38)(26,31,32,37)(27,34,35,36)(30,39,33,40)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8606493);