Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #2010 (order 128) · Inn(G) = C42 · Out(G) = D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C4 | C42 |
| Frattini, cosocle | C2×C8 | C22 |
| socle | C2 | #97 |
| prime radical | C22 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
23 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,7,4,9,11,15,5,8,10,16,12,6,13,14);(2,6,8,9)(3,10)(7,15,16,14)(11,13)Polycyclic presentation:
2^6|a^2=c,b^2=d,c^2=e,e^2=f,b^a=be,d^a=df,c^b=cf22 conjugacy classes — view the full character table.
| maximal subgroup of | #1355, #1356, C2×#347, … |
| maximal normal subgroup of | #1355, #1356, C2×#347, … |
| maximal quotient of | #1313, #1375, C2×#347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 28);