Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13479 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
788 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);(3,4,5,6)(7,8)(9,12)(10,20)(11,16)(13,21)(14,19)(15,24)(17,29)(18,27)(22,31)(23,34)(25,32)(26,36)(28,35)(30,33)(37,38);(7,9,12,8,13,16,28,18,14,17,29,19,27,35,11,21)(10,22,24,36,25,37,38,32,26,15,31,20,30,34,23,33);(4,6)(7,10,14,26)(8,15,19,22)(9,20,17,36)(11,24,28,31)(12,23,29,38)(13,30,27,25)(16,32,35,33)(18,37,21,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406532);