Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,6,7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,40)(38,39);(2,4)(3,7)(9,11,13,16)(10,14,15,12)(17,19,22,24)(18,20,21,23)(26,30)(27,33)(32,36)(35,39);(2,4)(6,8)(10,15)(12,14)(18,21)(19,24)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,4,5,2)(3,8,7,6)(9,12,13,14)(10,11,15,16)(17,21,22,18)(19,20,24,23)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4970663);