Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1530 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)(3,5,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,20)(19,22,23,24)(25,26,29,30)(27,33,35,31)(28,36,38,32)(34,39,37,40);(9,11,14,13)(10,15,12,16)(19,23)(22,24)(26,30)(31,33)(32,36)(39,40);(9,11,14,13)(10,16,12,15)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,36)(29,30)(32,38)(33,35)(34,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,15,13,16)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,15,12,16)(17,20,21,18)(19,24,23,22)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=fg,d^2=g,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3553759);