Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×#1390 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C43 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| socle | C23 | C2×#99 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4);(9,10)(11,15,16,13)(12,14);(5,6,7,8)(10,13)(14,15)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72278558, #72279017, C4×#12793 |
| maximal normal subgroup of | #72278558, #72279017, C4×#12793 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53556);