Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C8×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #120 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C2×B(4,𝔽2) | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1401 conjugacy classes of subgroups; only the 270 normal subgroups are shown.
This group has 1401 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,5,4,6,7,8);(13,16);(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,15)(14,16)128 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×B(4,𝔽2), C8×#1020⋊C2, C16×B(4,𝔽2), … |
| maximal normal subgroup of | SD16×B(4,𝔽2), C16×B(4,𝔽2), #72268313, … |
| maximal quotient of | C16×B(4,𝔽2), B(4,𝔽2)×#3633, B(4,𝔽2)×AGL(1,𝔽9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116000);