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×#46277 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6906 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)(9,15)(13,18)(16,21)(20,22)(24,28)(25,31)(26,33)(36,38);(8,12)(9,15)(10,17)(20,22)(23,24,27,28)(25,29,31,35)(26,34,33,30)(32,38,37,36);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,18)(16,20)(17,19)(21,22)(23,24)(25,29)(26,30)(27,28)(31,35)(32,36)(33,34)(37,38);(7,9,11,15)(8,13,12,18)(10,16,17,21)(14,20,19,22)(23,25)(24,29)(26,32)(27,31)(28,35)(30,36)(33,37)(34,38);(7,10,11,17)(8,14,12,19)(9,16,15,21)(13,20,18,22)(23,26,27,33)(24,30,28,34)(25,32,31,37)(29,36,35,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610179);