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×#45771 |
| 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 | |
6944 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)(23,24,29,34)(25,32,27,30)(26,31,33,28)(35,36,37,38);(8,12)(9,15)(10,17)(20,22)(23,25,29,27)(24,30,34,32)(26,37,33,35)(28,38,31,36);(3,4)(5,6)(7,8,11,12)(9,13,15,18)(10,14,17,19)(16,20,21,22)(23,26,29,33)(24,31,34,28)(25,35,27,37)(30,38,32,36);(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22)(23,27,29,25)(24,32,34,30)(26,37,33,35)(28,38,31,36);(7,10)(8,14)(9,16)(11,17)(12,19)(13,20)(15,21)(18,22)(23,28,29,31)(24,33,34,26)(25,36,27,38)(30,37,32,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609649);