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 | #50656 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(10,14)(11,17)(12,20)(22,24)(25,26)(27,28)(29,40)(30,39)(31,35)(32,38)(33,37)(34,36);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,34)(39,40);(2,5)(6,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,28,31,32)(26,27,35,38)(29,34,37,39)(30,40,36,33);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,29,31,37)(26,33,35,40)(27,36,38,30)(28,39,32,34);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27)(26,28)(29,36)(30,37)(31,38)(32,35)(33,39)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,30,31,36)(26,34,35,39)(27,37,38,29)(28,40,32,33)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8032869);