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 | |
4344 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)(11,14)(12,23)(13,15)(16,21)(17,20)(18,22)(19,24)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2)(3,6)(4,7)(5,8)(9,11,16,20)(10,17,21,14)(12,15,22,19)(13,23,24,18)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,15)(13,17)(14,24)(16,22)(19,20)(21,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13,16,24)(10,19,21,15)(11,18,20,23)(12,17,22,14)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(9,14,16,17)(10,20,21,11)(12,24,22,13)(15,18,19,23);(1,2)(3,5)(4,7)(6,8)(9,15,16,19)(10,13,21,24)(11,22,20,12)(14,18,17,23)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,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,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329242);