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×C22⋊#564 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6690 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)(7,8,12,15)(9,10,16,13)(11,14,19,21)(17,18,22,20)(23,24,29,27)(25,32,26,30)(28,35,36,31)(33,37,34,38);(9,16)(10,13)(11,19)(14,21)(23,25,29,26)(24,30,27,32)(28,33,36,34)(31,37,35,38);(3,4)(5,6)(7,9)(8,13)(10,15)(11,17)(12,16)(14,20)(18,21)(19,22)(23,26,29,25)(24,30,27,32)(28,34,36,33)(31,37,35,38);(7,10,12,13)(8,9,15,16)(11,18,19,20)(14,17,21,22)(23,27,29,24)(25,30,26,32)(28,35,36,31)(33,37,34,38);(7,11)(8,14)(9,17)(10,18)(12,19)(13,20)(15,21)(16,22)(23,28,29,36)(24,31,27,35)(25,33,26,34)(30,37,32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610024);