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 | |
4364 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)(11,20,15,17)(12,14,13,18)(21,23,22,24)(25,26,30,33)(27,29,32,39)(28,35,38,31)(34,40,36,37);(1,2)(3,5)(4,6)(7,8)(9,11,16,15)(10,17,19,20)(12,22,13,21)(14,23,18,24)(25,27,30,32)(26,29,33,39)(28,36,38,34)(31,40,35,37);(2,3)(6,7)(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,23,20,24)(25,28,30,38)(26,31,33,35)(27,34,32,36)(29,37,39,40);(4,8)(6,7)(9,13,16,12)(10,14,19,18)(11,22,15,21)(17,24,20,23)(25,27)(26,29)(28,34)(30,32)(31,37)(33,39)(35,40)(36,38);(1,3,5,2)(4,7,8,6)(9,14)(10,12)(11,23)(13,19)(15,24)(16,18)(17,22)(20,21)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,35)(39,40);(1,4)(2,6)(3,7)(5,8)(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,24,18,23)(25,29)(26,32)(27,33)(28,37)(30,39)(31,36)(34,35)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520206);