Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C8×#365 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #106 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#106 |
| prime radical | C23 | C4×SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
269 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 142 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);(10,14,15,20)(11,17)(12,19)(16,23,22,21)(18,24);(9,10,11,14,12,16,17,21,13,15,18,20,19,22,24,23)128 conjugacy classes — view the full character table.
| maximal subgroup of | C8×AGL(1,ℤ/16), C2×C8×#365, #72309408, … |
| maximal normal subgroup of | C8×AGL(1,ℤ/16), C2×C8×#365, #72309408, … |
| maximal quotient of | C8×#1368, C2×C8×#365 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36334);