Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C2×C8 |
| direct product | C8×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C82 | D4 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×C4⋊C8 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C82 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 214 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:
(1,2,3,5,4,6,7,8);(9,10,13,18,14,17,22,24,15,19,21,16,12,23,11,20);(9,11,12,21,15,22,14,13)(10,16,17,20,19,18,23,24)224 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#1369, C8×#1336, C8×#2171, … |
| maximal normal subgroup of | C8×#1369, C8×#1336, C8×#2171, … |
| maximal quotient of | C2×C8×#364 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36234);