Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#7187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(7,8,9,11,10,12,13,14)(15,16)(17,21,22,19)(18,20);(3,4,5,6)(16,19)(20,21)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238064, #72238066, #72238067, … |
| maximal normal subgroup of | #72238064, #72238066, #72238067, … |
| maximal quotient of | C2×#93464 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34103);