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 | |
4380 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)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(2,7)(3,4)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(28,36)(32,39)(35,40)(37,38);(1,3,6,4)(2,8,7,5)(25,27,30,33)(26,31,34,29)(28,35,36,40)(32,38,39,37);(1,4,6,3)(2,5,7,8)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,5)(2,3)(4,7)(6,8)(9,11)(10,17)(12,18)(13,15)(14,19)(16,23)(20,22)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,13,14,10)(11,17,19,15)(12,16,21,20)(18,22,24,23)(25,29,30,31)(26,33,34,27)(28,37,36,38)(32,40,39,35)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=g,f^2=gh,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, 8570170);