Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(10,14)(11,16)(19,23)(22,24)(25,26)(27,29)(28,31)(30,32);(1,2,5,3)(4,6,8,7)(9,10,13,14)(11,15,16,12)(18,21)(22,24)(26,27)(30,31);(9,11)(10,15)(12,14)(13,16)(17,18,20,21)(19,22,23,24)(25,27,29,26)(28,31,32,30);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,11,14,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215439);