Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1896 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,23,24,21)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,38)(37,39);(1,3,5,8)(2,6,7,4)(9,11,13,16)(10,14,12,15)(17,18,20,22)(19,21,24,23)(26,30)(27,34)(32,36)(35,37);(1,3)(2,4)(5,8)(6,7)(9,12,13,10)(11,14,16,15)(17,19)(18,21)(20,24)(22,23)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(1,4)(2,3)(5,6)(7,8)(25,27)(26,31)(28,37)(29,34)(30,33)(32,40)(35,38)(36,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,38)(26,32,30,36)(27,35,34,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3971178);