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 | |
4386 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,23)(19,22)(20,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,40)(36,39)(37,38);(1,3)(2,5)(4,8)(6,7)(10,15)(12,20)(16,22)(18,21)(25,27)(26,30)(28,34)(29,37)(31,33)(32,38)(35,39)(36,40);(10,15)(11,19)(17,24)(18,21)(26,28)(27,33)(32,36)(35,37);(9,11,14,19)(10,16,15,22)(12,18,20,21)(13,17,23,24)(25,28,31,26)(27,34,33,30)(29,36,39,32)(35,38,37,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,18)(13,22)(14,20)(15,24)(16,23)(19,21)(25,29,31,39)(26,32,28,36)(27,35,33,37)(30,40,34,38);(1,2)(3,5)(4,7)(6,8)(9,13,14,23)(10,18,15,21)(11,17,19,24)(12,16,20,22)(25,30,31,34)(26,33,28,27)(29,38,39,40)(32,35,36,37)Polycyclic presentation:
2^9|d^2=h,e^2=i,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, 9624714);