Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C4×#2127 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1539 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)(7,13,14,20)(8,16,18,11)(9,12)(10,15)(17,19);(5,7)(6,10)(8,17)(9,14)(11,20)(12,19)(13,16)(15,18);(7,14)(10,19)(13,20)(15,17)92 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#12753, C4×#33551, C4×#12707, … |
| maximal normal subgroup of | C4×#12753, C4×#33551, C4×#12707, … |
| maximal quotient of | AGL(1,𝔽5)×#2127, #72318038 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400203);