Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#62817 (order 512) · Inn(G) = C22 · Out(G) = C22×H(5,𝔽2) |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22 |
| commutator | C2 | C22×C8 |
| Frattini, cosocle | C2×C4 | C23 |
| socle | C22 | C22×C4 |
| prime radical | C23 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,13,12,14,15,16);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^6|a^2=e,c^2=d,e^2=f,b^a=bd40 conjugacy classes — view the full character table.
| maximal subgroup of | #1274, #1275, #1276, … |
| maximal normal subgroup of | #1274, #1275, #1276, … |
| maximal quotient of | #1744, C4×C22⋊C8, C4×C4⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 112);