Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142220 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 663 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3)(5,6,7,9)(8,10,12,11)(15,19);(1,2)(3,4)(6,9)(10,11)(14,17)(15,19)(18,20);(1,3,4,2)(6,10)(8,12)(9,11)(13,14)(15,20)(16,18)(17,19)Polycyclic presentation:
2^9|a^2=g,c^2=dg,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi,h^c=hi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52014);