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×#42168 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7074 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)(7,8,12,10)(9,13,15,17)(11,14,19,18)(16,20,21,22)(23,24,28,29)(25,30,32,27)(26,34,36,31)(33,35,38,37);(9,15)(11,19)(13,17)(14,18)(24,29)(25,32)(31,34)(33,38);(3,4,5,6)(7,9)(8,13)(10,17)(11,16)(12,15)(14,20)(18,22)(19,21)(23,25)(24,30)(26,33)(27,29)(28,32)(31,37)(34,35)(36,38);(7,10,12,8)(9,13,15,17)(11,18,19,14)(16,20,21,22)(23,26,28,36)(24,31,29,34)(25,33,32,38)(27,35,30,37);(7,11)(8,14)(9,16)(10,18)(12,19)(13,20)(15,21)(17,22)(23,27,28,30)(24,25,29,32)(26,35,36,37)(31,33,34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547836);