Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#50895 |
| 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 | |
6664 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)(9,13,16,21)(10,11,14,20)(17,22,19,18)(23,24)(25,28)(26,34)(27,30)(29,35)(31,33)(32,37)(36,38);(9,16)(10,14)(11,20)(13,21)(23,24,27,30)(25,28,33,31)(26,34,35,29)(32,37,38,36);(3,4,5,6)(7,9)(8,13)(10,19)(11,18)(12,16)(14,17)(15,21)(20,22)(23,24)(25,31)(26,29)(27,30)(28,33)(32,37)(34,35)(36,38);(7,10,12,14)(8,11,15,20)(9,17,16,19)(13,22,21,18)(23,25)(24,28)(26,32)(27,33)(29,36)(30,31)(34,37)(35,38);(7,11)(8,14)(9,18)(10,15)(12,20)(13,17)(16,22)(19,21)(23,26,27,35)(24,29,30,34)(25,32,33,38)(28,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613921);