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 | |
4695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,14)(11,17)(16,21)(18,23)(26,30)(27,34)(28,36)(38,39);(3,7)(6,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,33)(29,30)(32,34)(35,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,29,34)(26,31,30,32)(28,35,36,40)(33,38,37,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,32)(28,35)(29,34)(30,31)(33,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,26)(27,32)(28,33)(29,30)(31,34)(35,39)(36,37)(38,40);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,36)(26,33,30,37)(27,35,34,40)(31,38,32,39)Polycyclic presentation:
2^9|b^2=i,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034377);