Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,12)(13,16)(14,15)(19,23)(22,24)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(2,6)(3,8)(9,10,13,16)(11,12,15,14)(18,21)(22,24)(29,37)(32,39)(34,40)(36,38);(3,8)(4,7)(11,15)(12,14)(17,18,20,21)(19,22,23,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,3,5,8)(2,7,6,4)(9,11,13,15)(10,14,16,12)(25,28,30,31)(26,27,33,35)(29,36,37,38)(32,34,39,40);(1,4,5,7)(2,8,6,3)(9,12,13,14)(10,15,16,11)(17,19)(18,22)(20,23)(21,24)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=gh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5145396);