Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) = #3109 (order 128) · Inn(G) = #106 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #106 |
| commutator | C8 | C2×C4 |
| Frattini, cosocle | C2×C8 | C22 |
| prime radical | C22 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
21 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,6,7,12)(3,9)(4,11)(8,15,14,13)(10,16);(1,2,3,6,4,8,9,13,5,7,10,12,11,14,16,15)Polycyclic presentation:
2^6|a^2=d,b^2=c,c^2=e,e^2=f,b^a=bc,c^a=ce,e^a=ef,d^b=df16 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#365, #2155, #2180, … |
| maximal normal subgroup of | C2×#365, #2155, #2180, … |
| maximal quotient of | #1368, #1383, C2×#365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 46);