Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1732 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)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,21,23,24)(25,26,29,30)(27,33,35,31)(28,32,36,38)(34,40,37,39);(3,7)(6,8)(9,11)(10,14)(12,15)(13,16)(19,23)(21,24)(26,30)(28,36)(31,33)(34,37);(2,5)(6,8)(10,15)(11,16)(17,19,20,23)(18,21,22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(25,27,29,35)(26,31,30,33)(28,37,36,34)(32,40,38,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,36)(26,32,30,38)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fhi,c^2=hi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5627324);