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 | |
4377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,15)(11,16,18,12)(13,17,22,23)(19,21,24,20)(26,27)(29,37)(31,33)(35,38);(10,15)(11,18)(17,23)(19,24)(26,27)(31,33)(32,34)(39,40);(1,2,5,4)(3,6,7,8)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,39,38,40);(3,7)(6,8)(9,11)(10,16)(12,15)(13,19)(14,18)(17,21)(20,23)(22,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,39,38,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,11,15,18)(13,21,22,20)(17,24,23,19)(25,28)(26,31)(27,33)(29,38)(30,36)(32,40)(34,39)(35,37);(1,4,5,2)(3,8,7,6)(9,13,14,22)(10,17,15,23)(11,19,18,24)(12,20,16,21)(25,29,30,37)(26,32,27,34)(28,35,36,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10389629);