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 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,13,12,16)(14,17,22,24)(19,21,20,23)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,2)(3,7)(4,5)(6,8)(9,11,15,12)(10,16,18,13)(14,19,22,20)(17,23,24,21)(26,28)(27,33)(32,36)(35,39);(2,5)(6,7)(14,22)(17,24)(19,20)(21,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(3,8)(6,7)(9,12,15,11)(10,13,18,16)(14,20,22,19)(17,21,24,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,3)(2,6)(4,8)(5,7)(9,13)(10,12)(11,18)(14,21)(15,16)(17,20)(19,24)(22,23)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,17,18,24)(11,19,12,20)(13,21,16,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8254678);