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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1850 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
371 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 371 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(5,8)(9,10,13,17,14,11,18,20,12,16,21,24,22,19,15,23);(1,2)(3,5)(4,6)(7,8)(9,11,13,20,14,16,18,24,12,19,21,23,22,10,15,17);(1,3)(2,5)(4,7)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,e^b=eg,d^c=dg128 conjugacy classes — view the full character table.
| maximal subgroup of | C16×B(4,𝔽2), #72242819 |
| maximal normal subgroup of | C16×B(4,𝔽2) |
| maximal quotient of | #72242819 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43130);