Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C22⋊C4×#121 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(10,12)(13,14,15,16);(9,10)(11,12)(13,14)(15,16);(2,4)(6,8);(5,6,7,8);(1,2)(3,4)140 conjugacy classes — view the full character table.
| maximal subgroup of | #72233528, #72233554, #72233563, … |
| maximal normal subgroup of | #72233528, #72233554, #72233563, … |
| maximal quotient of | #72298143, #72298341, #72298397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426499);