Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#36595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(4,6)(7,8,11,9)(10,12,14,13)(15,16)(17,18)(19,21)(20,22);(7,8,11,9)(10,13,14,12)(15,17,19,22)(16,20,21,18);(3,4,5,6)(8,9)(12,13)(16,21)(17,22);(7,9,11,8)(10,12,14,13)(15,16)(17,20)(18,22)(19,21);(7,10,11,14)(8,12,9,13)(15,18,19,20)(16,17,21,22)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537503);