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 | |
4564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,14,16)(11,15,19,22)(12,21,23,17)(13,18,24,20);(2,7)(3,4)(9,11,14,19)(10,15,16,22)(12,20,23,18)(13,17,24,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,3,6,4)(2,8,7,5)(10,16)(11,19)(17,21)(18,20)(26,31)(27,35)(33,37)(34,36);(1,4,6,3)(2,5,7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,22)(17,24)(18,23)(25,27,30,35)(26,32,31,39)(28,36,38,34)(29,33,40,37);(1,5)(2,3)(4,7)(6,8)(9,12,14,23)(10,17,16,21)(11,20,19,18)(13,15,24,22)(25,28)(26,33)(27,36)(29,39)(30,38)(31,37)(32,40)(34,35);(9,13,14,24)(10,18,16,20)(11,17,19,21)(12,22,23,15)(25,29,30,40)(26,34,31,36)(27,37,35,33)(28,32,38,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8632050);