Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
| direct product | C2×#8202 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C4×D4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(8,12)(9,10)(13,14)(15,16,19,21,18,17,20,22);(4,6)(7,8,10,13,11,14,9,12)(16,17)(21,22);(15,17,19,22,18,16,20,21)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#220728, C2×#229439, C2×#183994, … |
| maximal normal subgroup of | C2×#220728, C2×#229439, C2×#183994, … |
| maximal quotient of | C2×#128714, C2×#128713, #72322749 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89622);