Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 10485760 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×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 | |
9051 conjugacy classes of subgroups; only the 750 normal subgroups are shown.
This group has 9051 conjugacy classes of subgroups — too many to draw. Only its 750 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,3)(5,6,8,7)(9,10,11,12)(14,16);(1,2,4,3)(5,6)(7,8)(9,10,11,12);(2,3)(6,7)(9,10)(11,12);(2,3)(6,7)(13,14)(15,16);(1,3,4,2)(5,7,8,6)(9,10)(11,12)Polycyclic presentation:
2^9|a^2=g,b^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4029418);