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×#13320 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(8,14)(9,19)(10,22)(11,12)(15,31)(16,23)(17,29)(18,30)(20,24)(21,27)(25,26)(28,33)(32,35)(34,37);(3,4)(5,6)(7,8,13,18)(9,15,21,34)(10,23,26,29)(11,17,12,16)(14,25,30,22)(19,35,27,33)(20,37,24,31)(28,36,32,38);(4,6)(7,9)(8,15)(10,24)(11,27)(12,19)(13,21)(14,28)(16,35)(17,33)(18,34)(20,26)(22,38)(23,31)(25,36)(29,37)(30,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406024);