Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#525 (order 128) · Inn(G) = C22 · Out(G) = C2×#41 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C22 |
| commutator | C2 | C2×C16 |
| socle | C22 | C2×C8 |
| prime radical | C23 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(5,6,7,10,8,11,13,16,9,12,14,17,15,18,19,20);(2,4)Polycyclic presentation:
2^6|a^2=d,d^2=e,e^2=f,b^a=bc40 conjugacy classes — view the full character table.
| maximal subgroup of | #1314, #1315, #1322, … |
| maximal normal subgroup of | #1314, #1315, #1322, … |
| maximal quotient of | #1314, #1322, D4⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 29);