Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24358 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,6)(7,8,9,10);(4,6)(12,15)(13,17)(16,18);(8,10)(11,12,14,16)(13,18,17,15);(4,6)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×SD16×#120, C2×#6437680, C2×#6573500, … |
| maximal normal subgroup of | C2×SD16×#120, C2×#6437680, C2×#6573500, … |
| maximal quotient of | C2×#359488 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344836);