Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#46132 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6982 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)(7,8,11,12)(9,15,17,13)(10,18,19,14)(16,20,22,21)(23,24,28,31)(25,26,32,29)(27,30,35,37)(33,34,38,36);(8,12)(10,19)(13,15)(16,22)(25,32)(26,29)(27,35)(30,37);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,17)(16,20)(18,19)(21,22)(23,25)(24,29)(26,31)(27,33)(28,32)(30,36)(34,37)(35,38);(7,9)(8,13)(10,16)(11,17)(12,15)(14,20)(18,21)(19,22)(23,26,28,29)(24,25,31,32)(27,34,35,36)(30,33,37,38);(7,10,11,19)(8,14,12,18)(9,16,17,22)(13,20,15,21)(23,27)(24,30)(25,33)(26,34)(28,35)(29,36)(31,37)(32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609993);