Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
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 | |
1953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,16)(13,15)(19,23)(22,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(3,8)(5,7)(9,11,13,16)(10,14,15,12)(18,21)(22,24)(26,30)(28,37)(31,33)(34,40);(10,15)(11,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5894846);