Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,4,3)(5,6,8,7)(10,12)(13,14,15,16);(2,3)(6,7)(13,14,15,16);(5,6,8,7)(13,14)(15,16);(9,10,11,12);(1,3,4,2)(5,7,8,6)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=gh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72226959, #72261412, #72261632, … |
| maximal normal subgroup of | #72226959, #72261412, #72261632, … |
| maximal quotient of | #72226904, #72298162, #72298198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 581889);