Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C4×#2408 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2408 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(6,7)(9,10)(11,12);(5,6,8,7)(13,14)(15,16);(10,12)(14,16);(5,7,8,6)152 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2408, #72233675, C4×D4×#120, … |
| maximal normal subgroup of | D4×#2408, #72233675, C4×D4×#120, … |
| maximal quotient of | #72214895, #72239140, #72239661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 422016);