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 | |
4496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,15)(11,16,19,14)(12,20,22,17)(18,23,21,24)(25,26)(27,37)(28,34)(29,30)(31,35)(32,36)(33,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,11,13,19)(10,14,15,16)(12,18,22,21)(17,23,20,24)(25,27)(26,33)(28,31)(29,38)(30,35)(32,39)(34,40)(36,37);(1,3)(2,5)(4,8)(6,7)(10,15)(14,16)(17,20)(23,24)(25,28,32,29)(26,34,36,30)(27,31,39,38)(33,40,37,35);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,17)(13,15)(16,19)(18,23)(20,22)(21,24)(25,29,32,28)(26,34,36,30)(27,38,39,31)(33,40,37,35);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,19)(14,15)(17,23)(18,22)(20,24)(25,30,32,34)(26,29,36,28)(27,35,39,40)(31,33,38,37);(1,2)(3,5)(4,7)(6,8)(9,12)(10,17)(11,18)(13,22)(14,23)(15,20)(16,24)(19,21)(25,31,32,38)(26,35,36,40)(27,28,39,29)(30,37,34,33)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9997638);