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 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,12,16,19)(13,21,23,17)(18,22,24,20)(25,26)(27,30)(28,35)(29,31)(32,36)(33,39)(34,37)(38,40);(2,3)(6,7)(10,15)(12,19)(17,21)(20,22)(25,27,29,34)(26,30,31,37)(28,33,36,40)(32,38,35,39);(1,3,5,2)(4,7,8,6)(10,15)(11,16)(17,21)(18,24)(26,31)(30,37)(32,35)(38,39);(1,4,5,8)(2,6,3,7)(9,11,14,16)(10,12,15,19)(13,18,23,24)(17,20,21,22)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,36)(39,40);(9,12)(10,16)(11,15)(13,22)(14,19)(17,18)(20,23)(21,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(9,13,14,23)(10,17,15,21)(11,18,16,24)(12,20,19,22)(25,28,29,36)(26,32,31,35)(27,33,34,40)(30,38,37,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9636422);