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 | |
4479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,4,8,5)(9,10,15,19)(11,16,21,23)(12,13,17,22)(14,18,20,24)(25,26,32,36)(27,31,29,34)(28,38,37,33)(30,35,39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,14)(13,18)(15,21)(17,20)(19,23)(22,24)(25,27,32,29)(26,31,36,34)(28,39,37,30)(33,35,38,40);(1,4,6,5)(2,3,7,8)(9,12,15,17)(10,13,19,22)(11,20,21,14)(16,24,23,18)(25,28,32,37)(26,33,36,38)(27,30,29,39)(31,40,34,35);(2,7)(4,5)(9,12)(10,13)(11,14)(15,17)(16,18)(19,22)(20,21)(23,24)(25,29,32,27)(26,34,36,31)(28,30,37,39)(33,40,38,35);(1,5,6,4)(2,3,7,8)(9,13)(10,17)(11,18)(12,19)(14,23)(15,22)(16,20)(21,24)(25,30)(26,35)(27,37)(28,29)(31,33)(32,39)(34,38)(36,40);(1,3)(2,5)(4,7)(6,8)(9,14)(10,18)(11,12)(13,16)(15,20)(17,21)(19,24)(22,23)(25,31,32,34)(26,27,36,29)(28,40,37,35)(30,38,39,33)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9048869);