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 | |
4235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(11,17)(16,19)(18,23)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,17,21)(12,16,20,19)(18,22,23,24)(26,31)(27,34)(33,36)(35,40);(2,5)(6,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(3,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(1,3)(2,6)(4,7)(5,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)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,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416897);