Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,20,24,22)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(1,3,6,8)(2,4,5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,22)(20,24)(21,23)(26,30)(27,33)(32,36)(35,39);(2,5)(3,8)(9,10,12,14)(11,13,16,15)(17,19,21,24)(18,22,23,20)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,5,6,2)(3,4,8,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=fhi,c^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4986162);