Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#37423 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6874 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)(8,12)(10,17)(13,18)(16,21)(23,24,28,29)(25,26,33,30)(27,35,36,31)(32,37,38,34);(8,12)(9,15)(14,19)(16,21)(24,29)(26,30)(31,35)(34,37);(3,4,5,6)(7,8)(9,13)(10,14)(11,12)(15,18)(16,20)(17,19)(21,22)(23,25)(24,30)(26,29)(27,32)(28,33)(31,37)(34,35)(36,38);(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22)(23,26,28,30)(24,25,29,33)(27,34,36,37)(31,32,35,38);(7,10,11,17)(8,14,12,19)(9,16,15,21)(13,20,18,22)(23,27,28,36)(24,31,29,35)(25,32,33,38)(26,34,30,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542050);