Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#29867 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,5,6)(7,8)(9,13)(10,14)(11,12)(15,21)(16,19)(17,22)(18,20)(23,24)(25,28)(26,29)(27,30)(31,38)(32,36)(33,35)(34,37);(8,12)(13,19)(14,20)(21,22)(23,25,26,32)(24,28,29,36)(27,31,34,33)(30,38,37,35);(4,6)(7,9,11,16)(8,13,12,19)(10,17,18,15)(14,22,20,21)(24,29)(25,31)(27,34)(28,35)(32,33)(36,38);(7,10,11,18)(8,14,12,20)(9,15,16,17)(13,21,19,22)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343337);