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) = C26×C4 |
| direct product | C2×#12573 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C23 | C2×#99 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8)(9,13)(10,17,18,12)(11,14)(15,21,22,19)(16,20);(3,4,5,6)(8,12)(13,19)(14,17)(20,21);(7,9,11,16)(8,13,14,20)(10,15,18,22)(12,19,17,21)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#492887, #72240229, #72240233, … |
| maximal normal subgroup of | C2×#492887, #72240229, #72240233, … |
| maximal quotient of | #72236729, #72278511, #72278558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400079);