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 | |
4442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,22)(23,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(26,31)(27,34)(33,36)(35,40);(1,2)(3,5)(4,6)(7,8)(10,15)(14,17)(16,20)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(9,12,13,21)(10,16,15,20)(11,18,19,24)(14,22,17,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,21)(23,24)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=ghi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104144);