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 | |
4613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,16)(11,15,18,12)(13,21,22,17)(19,20,24,23)(25,26)(27,36)(28,31)(29,39)(30,34)(32,35)(33,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,21)(22,24)(25,27,32,38)(26,33,35,36)(28,34,29,40)(30,31,37,39);(1,2,4,6)(3,5,8,7)(10,16)(11,18)(17,21)(19,24)(25,28,32,29)(26,31,35,39)(27,34,38,40)(30,33,37,36);(1,3)(2,5)(4,8)(6,7)(9,12,14,15)(10,11,16,18)(13,20,22,23)(17,19,21,24)(25,29,32,28)(26,31,35,39)(27,34,38,40)(30,36,37,33);(1,3)(2,5)(4,8)(6,7)(9,13,14,22)(10,17,16,21)(11,19,18,24)(12,20,15,23)(25,30,32,37)(26,34,35,40)(27,31,38,39)(28,33,29,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,23)(20,21)(22,24)(25,31)(26,28)(27,37)(29,35)(30,38)(32,39)(33,34)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=g,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8090921);