Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#2120 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C2×D4 | C25 |
| socle | C23 | C2×#99 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,11)(8,15,16,10)(9,12)(13,19,20,17)(14,18);(6,10)(11,17)(12,15)(18,19);(5,7,9,14)(6,11,12,18)(8,13,16,20)(10,17,15,19)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#493352, C2×#147782, C2×#146667, … |
| maximal normal subgroup of | C2×#493352, C2×#147782, C2×#146667, … |
| maximal quotient of | C2×#493352, #72234781, C2×#496228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529637);