Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(11,15)(14,16)(19,20)(23,24)(25,26,31,34)(27,29,28,32)(30,33,38,40)(35,37,36,39);(2,6)(3,8)(10,13)(14,16)(18,22)(23,24)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,20,24)(25,28,31,27)(26,29,34,32)(30,36,38,35)(33,37,40,39);(1,3)(2,7)(4,6)(5,8)(17,19,21,20)(18,23,22,24)(25,29)(26,28)(27,34)(30,37)(31,32)(33,36)(35,40)(38,39);(1,4,5,7)(2,3,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4431071);