Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22×C4 |
| direct product | C16×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C16 | D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,2,4,7,5,3,8,6);(1,3)(2,5)(4,6)(7,8)224 conjugacy classes — view the full character table.
| maximal subgroup of | C16×#453, #104×AGL(1,𝔽17) |
| maximal normal subgroup of | C16×#453 |
| maximal quotient of | #104×AGL(1,𝔽17) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35836);