Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C24×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | D4 |
| commutator | C4 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C25 | C22⋊C4 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
26434 conjugacy classes of subgroups; only the 8034 normal subgroups are shown.
This group has 26434 conjugacy classes of subgroups — too many to draw. Only its 8034 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 — 6 generators on 16 points:
(1,2);(3,4);(5,6);(7,8);(9,10,12,15,13,11,16,14);(9,11)(10,13)(12,14)(15,16)224 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#10923, C2×#123640, C22×#20606, … |
| maximal normal subgroup of | C22×#10923, C2×#123640, C22×#20606, … |
| maximal quotient of | C22×#21415, C24×D4⋊C8, C23×#1896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481237);