Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13415 |
| 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 | |
1194 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)(7,8,14,20)(9,12,23,30)(10,21,27,37)(11,19,13,17)(15,18,28,22)(16,32,34,26)(24,29,33,38)(25,35,31,36);(7,9,14,23)(8,15,20,28)(10,24,27,33)(11,18,13,22)(12,17,30,19)(16,25,34,31)(21,32,37,26)(29,36,38,35);(4,6)(7,10,11,25,14,27,13,31)(8,16,17,33,20,34,19,24)(9,21,22,36,23,37,18,35)(12,29,28,26,30,38,15,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406273);