Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2019 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)(4,5)(6,8)(9,10,13,14)(11,12,15,16)(18,21)(19,23)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,39,38,40);(1,2)(3,7)(4,5)(6,8)(9,11,13,15)(10,12,14,16)(19,23)(22,24)(26,30)(28,37)(31,33)(34,38);(2,5)(6,7)(10,14)(11,15)(17,18,20,21)(19,22,23,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,35)(26,31,30,33)(28,38,37,34)(32,40,36,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,28,29,37)(26,32,30,36)(27,34,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fghi,c^2=gh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4100723);