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 | |
4555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,26)(27,29)(28,33)(30,31)(32,34)(35,37)(36,38)(39,40);(1,2)(3,7)(4,5)(6,8)(10,13)(15,17)(16,21)(22,23)(26,31)(28,36)(29,32)(35,40);(2,5)(6,7)(11,18)(12,20)(15,17)(16,21)(26,31)(27,34)(28,36)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,17,18,15)(12,21,20,16)(19,23,24,22)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)Polycyclic presentation:
2^9|a^2=gh,d^2=h,e^2=h,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, 8666462);