Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#2120 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #99 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1502 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)(7,11)(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 | #72280656, C4×#12711, #72282049, … |
| maximal normal subgroup of | #72280656, C4×#12711, #72282049, … |
| maximal quotient of | #72280760, C4×#12793, #72283068 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400183);