Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1876 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,20)(19,22,24,23)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,20,23)(26,30)(27,33)(28,36)(38,40);(2,3)(6,7)(10,15)(11,16)(18,20)(22,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,20,21,18)(19,23,24,22)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fgi,d^2=fgh,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, 4974991);