Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D8×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D8×D16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 4 generators on 24 points:
(1,2,3,4,5,6,7,8);(2,8)(3,7)(4,6);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24);(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)77 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D8×D16, D162, #72226906, … |
| maximal normal subgroup of | C2×D8×D16, D162, #72226906, … |
| maximal quotient of | #72225561, #72225563, #72225565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399715);