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 | #45623 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10)(11,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26,31,33)(27,29,30,32)(28,36,39,34)(35,40,37,38);(1,2,6,5)(3,8,4,7)(9,11,14,18)(10,13,15,17)(12,20,22,19)(16,21,24,23)(25,27,31,30)(26,32,33,29)(28,37,39,35)(34,38,36,40);(1,3,6,4)(2,7,5,8)(10,15)(11,18)(16,24)(19,20)(26,33)(29,32)(34,36)(38,40);(1,4,6,3)(2,8,5,7)(9,12)(10,16)(11,19)(13,23)(14,22)(15,24)(17,21)(18,20)(25,28,31,39)(26,34,33,36)(27,35,30,37)(29,38,32,40);(1,5,6,2)(3,8,4,7)(9,11)(10,17)(12,19)(13,15)(14,18)(16,21)(20,22)(23,24)(25,29,31,32)(26,27,33,30)(28,38,39,40)(34,35,36,37);(9,13,14,17)(10,11,15,18)(12,21,22,23)(16,20,24,19)(25,30,31,27)(26,32,33,29)(28,35,39,37)(34,38,36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8546990);