Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10084 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,13,18)(9,19,10,21)(11,17,12,16)(14,22,15,20)(23,24,28,26)(25,31,34,38)(27,35,37,30)(29,36,32,33);(3,4)(5,6)(7,9)(8,14)(10,13)(11,22)(12,20)(15,18)(16,21)(17,19)(23,25,28,34)(24,29,26,32)(27,36,37,33)(30,38,35,31);(4,6)(7,10,13,9)(8,15,18,14)(11,22,12,20)(16,21,17,19)(23,26,28,24)(25,32,34,29)(27,30,37,35)(31,36,38,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140832);