Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,15,13)(11,12,16,14)(19,23)(22,24)(28,32)(30,31);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,13)(15,16)(18,21)(22,24)(25,26,29,27)(28,30,32,31);(1,2)(3,5)(4,6)(7,8)(9,12,15,14)(10,16,13,11)(17,18)(19,22)(20,21)(23,24)(25,27,29,26)(28,31,32,30);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,12,16,14)(25,28)(26,30)(27,31)(29,32);(1,3,4,8)(2,5,6,7)(9,14,15,12)(10,16,13,11)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fgi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4364132);