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 | |
4285 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)(5,8)(6,7)(9,10)(11,16)(12,14)(13,18)(15,17)(19,24)(20,23)(21,22)(25,26,32,30)(27,36,38,33)(28,34,29,39)(31,35,37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,19)(14,22)(15,20)(17,23)(18,24)(25,27)(26,33)(28,31)(29,37)(30,36)(32,38)(34,40)(35,39);(1,4)(2,7)(3,8)(5,6)(10,17)(11,20)(12,19)(18,22)(25,28,32,29)(26,34,30,39)(27,31,38,37)(33,40,36,35);(2,8)(4,6)(9,12,15,19)(10,14,17,24)(11,21,20,13)(16,22,23,18)(25,29,32,28)(26,34,30,39)(27,37,38,31)(33,40,36,35);(1,4)(2,3)(5,6)(7,8)(9,13,15,21)(10,18,17,22)(11,12,20,19)(14,23,24,16)(25,30,32,26)(27,36,38,33)(28,39,29,34)(31,35,37,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,19)(11,22)(12,17)(13,16)(15,24)(18,20)(21,23)(25,31,32,37)(26,35,30,40)(27,28,38,29)(33,39,36,34)Polycyclic presentation:
2^9|a^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=ghi,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, 10103111);