Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#51979 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6984 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);(3,4,5,6)(7,8)(9,13)(10,14)(11,12)(15,17)(16,21)(18,22)(19,20)(24,25)(27,35)(29,31)(33,38);(8,11)(9,15)(14,19)(16,18)(23,24,28,25)(26,31,34,29)(27,30,35,32)(33,37,38,36);(3,4)(5,6)(7,9,12,15)(8,13,11,17)(10,18,20,16)(14,22,19,21)(23,25,28,24)(26,31,34,29)(27,32,35,30)(33,37,38,36);(7,10,12,20)(8,14,11,19)(9,16,15,18)(13,21,17,22)(23,26,28,34)(24,29,25,31)(27,33,35,38)(30,36,32,37);(7,11,12,8)(9,17,15,13)(10,19,20,14)(16,22,18,21)(23,27)(24,30)(25,32)(26,33)(28,35)(29,36)(31,37)(34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615224);