Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#13308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,13,19)(9,20,23,36)(10,24,28,35)(11,18,12,17)(14,30,33,21)(15,29,34,22)(16,26,31,27)(25,37,38,32);(3,4)(5,6)(7,9)(8,14)(10,25)(11,22)(12,29)(13,23)(15,31)(16,34)(17,32)(18,37)(19,33)(20,24)(21,27)(26,30)(28,38)(35,36);(4,6)(7,9)(8,15)(10,21)(11,29)(12,22)(13,23)(14,31)(16,33)(17,36)(18,20)(19,34)(24,37)(25,27)(26,38)(28,30)(32,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406006);