Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#2197 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×B(4,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1316 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)(7,13)(8,11)(9,12)(10,14)(15,19)(16,18)(17,20);(5,7,9,15)(6,10,12,17)(8,14,16,20)(11,13,18,19);(8,16)(14,20)80 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#33551, AGL(1,𝔽5)×#2197, C4×#73030, … |
| maximal normal subgroup of | C4×#33551, #72278353, #72278484, … |
| maximal quotient of | AGL(1,𝔽5)×#2197 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402938);