Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C22×#3003 (order 512) · Inn(G) = D4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4 |
| commutator | C4 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| socle | C22 | C2×D4 |
| prime radical | C2×D8 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4);(5,6)(7,9)(8,11)(10,12);(6,9)(7,12)(10,11)28 conjugacy classes — view the full character table.
| maximal subgroup of | #1331, #1333, C8×D8, … |
| maximal normal subgroup of | #1331, #1333, C8×D8, … |
| maximal quotient of | C8×D8, #1581, #1589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 118);