Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#46817 |
| 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 | |
7210 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)(7,8,12,9)(10,13,17,15)(11,16,19,14)(18,21,22,20)(23,24)(25,30)(26,31)(27,35)(28,29)(32,36)(33,34)(37,38);(8,9)(10,17)(11,19)(20,21)(24,29)(26,30)(32,35)(34,37);(3,4)(5,6)(7,9,12,8)(10,13,17,15)(11,16,19,14)(18,20,22,21)(23,25,28,31)(24,30,29,26)(27,33,36,38)(32,37,35,34);(7,10,12,17)(8,13,9,15)(11,18,19,22)(14,20,16,21)(23,26,28,30)(24,31,29,25)(27,34,36,37)(32,38,35,33);(7,11)(8,14)(9,16)(10,18)(12,19)(13,20)(15,21)(17,22)(23,27)(24,32)(25,33)(26,34)(28,36)(29,35)(30,37)(31,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7607882);