Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#36699 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7002 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);(3,4,5,6)(7,8)(9,13)(10,14)(11,12)(15,18)(16,20)(17,19)(21,22)(23,24,27,30)(25,28,32,36)(26,29,34,38)(31,35,33,37);(8,12)(13,18)(14,19)(20,22)(23,25,27,32)(24,28,30,36)(26,33,34,31)(29,37,38,35);(4,6)(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22)(23,26)(24,29)(25,31)(27,34)(28,35)(30,38)(32,33)(36,37);(9,15)(13,18)(16,21)(20,22)(24,30)(26,34)(28,36)(31,33);(7,10,11,17)(8,14,12,19)(9,16,15,21)(13,20,18,22)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540606);