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 | |
1906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(10,12)(13,16)(17,18,22,21)(19,23,24,20)(26,30)(27,33)(32,37)(35,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,13,15,16)(17,19)(18,20)(21,23)(22,24)(28,36)(32,37)(35,39)(38,40);(3,7)(6,8)(9,11,14,15)(10,13,12,16)(17,20,22,23)(18,19,21,24)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,13,15,16)(17,21,22,18)(19,23,24,20)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,2)(3,8)(4,5)(6,7)(9,13)(10,15)(11,12)(14,16)(17,19)(18,23)(20,21)(22,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=ghi,d^2=f,e^2=g,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, 5660779);