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 | |
4379 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)(11,19)(12,17)(13,15)(14,18)(16,20)(21,22)(23,24)(25,26)(27,33)(28,38)(29,35)(30,36)(31,32)(34,39)(37,40);(1,2,5,4)(3,6,7,8)(9,11)(10,16)(12,14)(13,19)(15,20)(17,23)(18,22)(21,24)(25,27,32,29)(26,33,31,35)(28,30,39,37)(34,40,38,36);(2,4)(3,7)(9,12,15,21)(10,17,13,22)(11,14,20,24)(16,23,19,18)(25,28,32,39)(26,34,31,38)(27,37,29,30)(33,36,35,40);(1,3,5,7)(2,6,4,8)(9,13,15,10)(11,19,20,16)(12,17,21,22)(14,23,24,18)(25,29,32,27)(26,35,31,33)(28,37,39,30)(34,36,38,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,14)(13,22)(15,21)(16,23)(18,19)(20,24)(25,30,32,37)(26,36,31,40)(27,28,29,39)(33,38,35,34);(1,4,5,2)(3,8,7,6)(9,14)(10,18)(11,12)(13,23)(15,24)(16,22)(17,19)(20,21)(25,31,32,26)(27,35,29,33)(28,34,39,38)(30,40,37,36)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=gi,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8939644);