Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
1708 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)(9,10,14,12)(11,13,16,15)(19,23)(21,24)(26,30)(27,33)(28,36)(39,40);(4,8)(6,7)(11,16)(13,15)(25,26,29,30)(27,34,33,31)(28,37,36,32)(35,39,38,40);(1,2,5,3)(4,6,8,7)(9,11,14,16)(10,15,12,13)(17,18,20,22)(19,21,23,24)(25,26)(27,31)(28,37)(29,30)(32,36)(33,34)(35,40)(38,39);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,19)(18,21)(20,23)(22,24)(25,27,29,33)(26,31,30,34)(28,38,36,35)(32,40,37,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(25,28)(26,32)(27,35)(29,36)(30,37)(31,39)(33,38)(34,40)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=hi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3971196);