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 | |
4693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,12)(13,21)(14,15)(16,17)(18,22)(19,23)(20,24)(25,26)(27,33)(28,36)(29,30)(31,34)(32,38)(35,39)(37,40);(1,2,4,6)(3,7,8,5)(10,15)(12,16)(13,22)(19,24)(26,30)(31,33)(32,36)(39,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,17)(18,21)(19,24)(27,34)(31,33)(35,37)(39,40);(3,8)(5,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,27,29,34)(26,31,30,33)(28,37,38,35)(32,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,29,38)(26,32,30,36)(27,35,34,37)(31,39,33,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8751077);