Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37558 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(10,18)(13,20)(16,22)(19,21)(23,24)(25,31)(26,29)(27,30)(28,33)(32,37)(34,36)(35,38);(7,8,11,14)(9,12,17,15)(10,13,18,20)(16,19,22,21)(23,24,27,30)(25,28,33,31)(26,29,34,36)(32,35,38,37);(3,4,5,6)(7,8)(9,15)(10,13)(11,14)(12,17)(16,21)(18,20)(19,22)(23,24)(25,28)(26,29)(27,30)(31,33)(32,35)(34,36)(37,38);(7,9)(8,12)(10,16)(11,17)(13,19)(14,15)(18,22)(20,21)(23,25,27,33)(24,28,30,31)(26,32,34,38)(29,35,36,37);(7,10)(8,13)(9,16)(11,18)(12,19)(14,20)(15,21)(17,22)(23,26,27,34)(24,29,30,36)(25,32,33,38)(28,35,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541038);