Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,13)(11,15)(18,22)(20,24)(25,26,29,30)(27,33,34,31)(28,36,38,32)(35,39,37,40);(1,2,5,4)(3,6,7,8)(11,15)(14,16)(18,22)(19,23)(26,30)(27,34)(32,36)(35,37);(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24)(25,27,29,34)(26,31,30,33)(28,37,38,35)(32,40,36,39);(1,4,5,2)(3,8,7,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28,29,38)(26,32,30,36)(27,35,34,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=f,b^2=fghi,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2629396);