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 | |
4311 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)(3,4)(5,6)(7,8)(26,30)(28,35)(31,36)(34,39);(2,6)(4,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(27,33)(31,36)(34,39)(38,40);(2,6)(3,8)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,5,8)(2,7,6,4)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4,5,7)(2,8,6,3)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8725327);