Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,6,8,7)(9,10,13,11)(12,15,16,14)(17,18,20,21)(19,23,24,22)(25,26,31,33)(27,34,28,30)(29,37,39,32)(35,38,36,40);(2,3)(4,8)(9,10)(11,13)(12,15)(14,16)(18,21)(19,24)(25,27,31,28)(26,30,33,34)(29,35,39,36)(32,38,37,40);(2,3)(6,7)(10,11)(14,15)(17,19,20,24)(18,22,21,23)(25,28,31,27)(26,30,33,34)(29,35,39,36)(32,40,37,38);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,24)(21,23)(25,29,31,39)(26,32,33,37)(27,35,28,36)(30,38,34,40);(1,4,5,8)(2,6,3,7)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,30)(26,27)(28,33)(29,38)(31,34)(32,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=i,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4431367);