Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
804 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);(3,4,6,5)(7,8)(9,17)(10,22)(11,27)(12,29)(13,18)(14,20)(15,24)(16,28)(19,33)(21,35)(23,37)(25,34)(26,38)(30,32)(31,36);(3,5,6,4)(7,9,11,20,13,21,28,12)(8,14,16,17,18,29,27,35)(10,19,23,36,26,24,25,30)(15,22,31,34,33,38,32,37);(7,10,11,23,13,26,28,25)(8,15,16,31,18,33,27,32)(9,19,20,36,21,24,12,30)(14,22,17,34,29,38,35,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402064);