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 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,19)(11,20,22,17)(12,15,13,23)(14,18,21,24)(25,26)(27,31)(28,37)(29,32)(30,34)(33,36)(35,40)(38,39);(1,2)(3,6)(4,7)(5,8)(9,11,16,22)(10,17,19,20)(12,21,13,14)(15,18,23,24)(25,27)(26,31)(28,38)(29,35)(30,36)(32,40)(33,34)(37,39);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,15,19,23)(11,21,22,14)(17,18,20,24)(25,28,30,29)(26,32,34,37)(27,35,36,38)(31,39,33,40);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,15,19,23)(11,14,22,21)(17,18,20,24)(25,26)(27,33)(28,32)(29,37)(30,34)(31,36)(35,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,14)(10,18)(11,12)(13,22)(15,20)(16,21)(17,23)(19,24)(25,27)(26,33)(28,35)(29,38)(30,36)(31,34)(32,40)(37,39);(1,2)(3,5)(4,7)(6,8)(9,15)(10,13)(11,18)(12,19)(14,17)(16,23)(20,21)(22,24)(25,29,30,28)(26,32,34,37)(27,35,36,38)(31,40,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8566616);