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×#10083 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,12,16)(9,17,19,22)(10,15,11,14)(13,20,21,18)(23,24,27,30)(25,31,33,38)(26,34,36,29)(28,35,37,32);(3,4)(5,6)(7,9)(8,13)(10,20)(11,18)(12,19)(14,22)(15,17)(16,21)(23,25,27,33)(24,28,30,37)(26,35,36,32)(29,38,34,31);(4,6)(7,9)(8,13)(10,18)(11,20)(12,19)(14,17)(15,22)(16,21)(26,36)(29,34)(31,38)(32,35)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140829);