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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1719 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)(3,7)(4,8)(5,6)(9,10)(11,12)(13,15)(14,16)(17,18,21,22)(19,20,23,24)(27,33)(28,35)(31,36)(32,37);(2,6)(3,8)(9,11)(10,15)(12,16)(13,14)(17,19,21,23)(18,20,22,24)(26,30)(31,36)(32,37)(38,40);(3,8)(4,7)(10,16)(11,13)(18,22)(19,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4,5,7)(2,8,6,3)(9,13,14,11)(10,12,16,15)(17,20)(18,19)(21,24)(22,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=fhi,e^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, 5030973);