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 | |
4360 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)(11,18)(12,15)(13,16)(14,20)(17,22)(19,23)(21,24)(25,26,31,36)(27,32,38,40)(28,29,34,33)(30,39,37,35);(1,2)(3,7)(4,5)(6,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27,31,38)(26,32,36,40)(28,37,34,30)(29,35,33,39);(2,5)(6,7)(9,10,15,12)(11,18,13,16)(14,17,22,20)(19,24,21,23)(25,28)(26,33)(27,37)(29,36)(30,38)(31,34)(32,39)(35,40);(2,5)(6,7)(9,12,15,10)(11,18,13,16)(14,20,22,17)(19,24,21,23)(25,28)(26,29)(27,37)(30,38)(31,34)(32,35)(33,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,16,12,18)(14,21,22,19)(17,23,20,24)(25,29)(26,34)(27,35)(28,36)(30,32)(31,33)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30,31,37)(26,35,36,39)(27,28,38,34)(29,32,33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10229903);