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 | |
4378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(3,8)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,26,31,35)(27,36,37,32)(28,33,39,38)(29,30,34,40);(1,2,5,6)(3,4,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(27,37)(28,39)(32,36)(33,38);(1,3)(2,4)(5,8)(6,7)(25,27,31,37)(26,32,35,36)(28,34,39,29)(30,38,40,33);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,28)(26,33)(27,34)(29,37)(30,32)(31,39)(35,38)(36,40);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,29,31,34)(26,30,35,40)(27,28,37,39)(32,38,36,33);(1,4,5,7)(2,8,6,3)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,30)(26,34)(27,38)(28,36)(29,35)(31,40)(32,39)(33,37)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8344027);