Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×#1335 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D8 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | D4⋊C8 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2);(3,4);(5,6,9,14,11,15,20,13,12,16,8,19,10,7,17,18);(5,7)(6,11)(8,13)(9,18)(10,16)(12,15)(14,20)(17,19)176 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#67992, C2×#143196, #72215347, … |
| maximal normal subgroup of | C2×#143196, #72215347, C2×#128570, … |
| maximal quotient of | C22×#7114, C2×D3×#1335, C2×C4×#1335 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388264);