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×#42978 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6814 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)(7,8,12,11)(9,15,17,13)(10,18,20,14)(16,21,19,22)(23,24,28,31)(25,29,33,27)(26,30,36,38)(32,37,34,35);(8,11)(13,15)(14,18)(21,22)(23,25,28,33)(24,29,31,27)(26,32,36,34)(30,37,38,35);(3,4)(5,6)(7,9,12,17)(8,13,11,15)(10,19,20,16)(14,22,18,21)(23,25)(24,29)(26,34)(27,31)(28,33)(30,35)(32,36)(37,38);(7,10,12,20)(8,14,11,18)(9,16,17,19)(13,21,15,22)(23,26,28,36)(24,30,31,38)(25,32,33,34)(27,35,29,37);(7,11,12,8)(9,15,17,13)(10,18,20,14)(16,22,19,21)(23,27,28,29)(24,25,31,33)(26,35,36,37)(30,32,38,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548899);