Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1779 |
| prime radical | C2×H(5,𝔽2) | C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
783 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 783 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,3)(2,5)(4,8)(6,7);(1,2)(3,5)(4,6)(7,8)Polycyclic presentation:
2^9|a^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^b=eg,f^b=fg,d^c=dg128 conjugacy classes — view the full character table.
| maximal subgroup of | C16×B(4,𝔽2), #72242813, #72287551, … |
| maximal normal subgroup of | C16×B(4,𝔽2), #72287551, #72320508, … |
| maximal quotient of | #72242813, #72287798, #72287804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36763);