Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C213 |
| direct product | C22×#1790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
20158 conjugacy classes of subgroups; only the 1858 normal subgroups are shown.
This group has 20158 conjugacy classes of subgroups — too many to draw. Only its 1858 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) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(5,6)(7,8)(9,10)(11,15,16,13)(12,14);(9,11)(10,13)(12,16)(14,15);(6,8)(9,10)(11,13)(12,14)(15,16)128 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#32898, #72219233, #72225622, … |
| maximal normal subgroup of | C22×#32898, #72219233, #72225622, … |
| maximal quotient of | C22×#8534, #72233659, C2×#185258, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277926);