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 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(10,14)(12,16)(19,23)(22,24)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(2,3)(6,7)(10,14)(11,15)(18,21)(22,24)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(1,2,5,3)(4,6,8,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fi,e^2=f,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, 4422775);