Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#12958 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4,6,5)(7,8)(9,11)(10,12)(13,14);(3,5,6,4)(8,11)(12,14);(9,13)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#512030, C2×#495771, C2×#512142, … |
| maximal normal subgroup of | C2×#512030, C2×#495771, C2×#512142, … |
| maximal quotient of | #72261418, #72281127, #72282512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401539);