Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C43 |
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 | |
6644 conjugacy classes of subgroups; only the 1002 normal subgroups are shown.
This group has 6644 conjugacy classes of subgroups — too many to draw. Only its 1002 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,8)(10,12)(13,14)(15,19)(16,18)(17,20);(5,6,7,8)(13,14,16,18)(15,17,20,19);(9,10,11,12)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20);(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=h,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, 4836121);