Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,13,16,15)(17,18,20,22)(19,23,24,21)(26,30)(31,33)(32,37)(39,40);(3,8)(5,7)(11,16)(13,15)(25,26,29,30)(27,33,35,31)(28,32,38,37)(34,40,36,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,15,12,13)(25,27,29,35)(26,31,30,33)(28,36,38,34)(32,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,10)(11,13,16,15)(17,19,20,24)(18,21,22,23)(25,26)(27,31)(28,37)(29,30)(32,38)(33,35)(34,40)(36,39);(1,3)(2,5)(4,8)(6,7)(9,13)(10,11)(12,16)(14,15)(25,28,29,38)(26,32,30,37)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=h,d^2=fgi,e^2=f,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, 4010702);