Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #394058 (order 512) · Inn(G) = C22 · Out(G) = #2021 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C22 |
| commutator | C2 | C4×C8 |
| socle, prime radical | C23 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,4)(9,10,11,13,12,14,15,16);(2,4)(5,6,7,8)Polycyclic presentation:
2^6|a^2=d,b^2=e,d^2=f,b^a=bc40 conjugacy classes — view the full character table.
| maximal subgroup of | #1270, #1271, #1272, … |
| maximal normal subgroup of | #1270, #1271, #1272, … |
| maximal quotient of | #1273, #1274, #1275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 17);