Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
2104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,26)(27,32)(28,29)(30,31);(2,3)(6,7)(17,19,21,20)(18,22,23,24)(26,31)(27,28);(2,3)(6,7)(10,14)(11,16)(18,23)(22,24)(25,27,30,28)(26,32,31,29);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,28,30,27)(26,29,31,32);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,11,14,16)(17,20,21,19)(18,24,23,22)(25,29)(26,27)(28,31)(30,32)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836796);