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×#42316 |
| 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 | |
6728 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)(23,24)(25,31)(26,30)(27,28)(29,33)(32,37)(34,35)(36,38);(8,12)(10,18)(13,15)(16,22)(24,28)(29,31)(30,35)(36,37);(3,4)(5,6)(7,8,11,12)(9,15,17,13)(10,14,18,19)(16,21,22,20)(23,24)(25,29)(26,30)(27,28)(31,33)(32,36)(34,35)(37,38);(7,9,11,17)(8,13,12,15)(10,16,18,22)(14,20,19,21)(23,25)(24,29)(26,32)(27,33)(28,31)(30,36)(34,38)(35,37);(7,10)(8,14)(9,16)(11,18)(12,19)(13,20)(15,21)(17,22)(23,26,27,34)(24,30,28,35)(25,32,33,38)(29,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548013);