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 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,15,18)(11,19,21,14)(12,22,24,16)(13,23,20,17);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,21)(16,23)(17,22)(18,19)(20,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,12,15,24)(10,16,18,22)(11,13,21,20)(14,23,19,17)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,40,39);(1,3,5,8)(2,6,7,4)(9,12)(10,16)(11,20)(13,21)(14,17)(15,24)(18,22)(19,23)(25,28,30,38)(26,32,34,40)(27,29,36,37)(31,39,35,33);(1,2)(3,6)(4,8)(5,7)(9,13,15,20)(10,17,18,23)(11,12,21,24)(14,22,19,16)(25,26)(27,31)(28,32)(29,39)(30,34)(33,37)(35,36)(38,40);(1,4,5,6)(2,3,7,8)(9,14)(10,11)(12,23)(13,16)(15,19)(17,24)(18,21)(20,22)(25,29)(26,33)(27,28)(30,37)(31,40)(32,35)(34,39)(36,38)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8626145);