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 | C2×#14185 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
8642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,6,5)(8,10)(16,18);(7,8)(9,10)(12,14)(16,18);(11,12)(13,14)(15,16,17,18);(3,5,6,4)(7,8)(9,10)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#786834, C2×#773154, C2×#1418370, … |
| maximal normal subgroup of | C2×#786834, C2×#773154, C2×#1418370, … |
| maximal quotient of | C2×#154655, C2×#154494, C2×#153868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421866);