Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = D4×AGL(1,ℤ/8) (order 256) · Inn(G) = #121 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,15);(1,2,7,14,5,9,4,6)(3,8,10,15,12,16,11,13);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,16)(14,15)Polycyclic presentation:
2^6|b^2=df,d^2=f,e^2=f,b^a=bd,c^a=ce,d^a=df,e^a=ef,e^c=ef16 conjugacy classes — view the full character table.
| maximal subgroup of | #2192, #2193, #2221, … |
| maximal normal subgroup of | #2192, #2193, #2221, … |
| maximal quotient of | #1667, #1668, #1675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 153);