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 | |
4675 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)(3,5,7,8)(9,10)(11,16)(12,18)(13,14)(15,17)(19,23)(20,24)(21,22)(26,31)(27,34)(33,37)(35,40);(3,7)(5,8)(9,11)(10,16)(12,19)(13,15)(14,17)(18,23)(20,21)(22,24)(25,26,30,31)(27,32,34,28)(29,37,38,33)(35,36,40,39);(1,2)(3,5)(4,6)(7,8)(10,17)(11,13)(12,20)(22,23)(27,34)(28,32)(35,40)(36,39);(9,12,15,20)(10,18,17,24)(11,19,13,21)(14,22,16,23)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,13,15,11)(10,14,17,16)(12,21,20,19)(18,22,24,23)(25,28,30,32)(26,27,31,34)(29,36,38,39)(33,35,37,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,11)(12,22)(13,17)(15,16)(18,19)(20,23)(21,24)(25,29,30,38)(26,33,31,37)(27,35,34,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=hi,b^2=g,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8707443);