Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#42304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(24,26)(25,31)(30,34)(32,37);(8,12)(10,18)(13,15)(16,22)(23,24,28,26)(25,29,31,33)(27,34,35,30)(32,38,37,36);(3,4)(5,6)(7,8,11,12)(9,15,17,13)(10,14,18,19)(16,21,22,20)(23,25)(24,29)(26,33)(27,32)(28,31)(30,36)(34,38)(35,37);(7,9,11,17)(8,13,12,15)(10,16,18,22)(14,20,19,21)(23,26,28,24)(25,29,31,33)(27,34,35,30)(32,36,37,38);(7,10)(8,14)(9,16)(11,18)(12,19)(13,20)(15,21)(17,22)(23,27,28,35)(24,30,26,34)(25,32,31,37)(29,36,33,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548021);