Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = #612424 (order 512) · Inn(G) = C2×D4 · Out(G) = #120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| prime radical | C23 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
52 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)(12,14,16,15);(2,5)(6,8)(10,13)(14,15);(1,3,4,7)(2,6,5,8)(10,14)(12,16)(13,15)Polycyclic presentation:
2^6|a^2=f,c^2=df,e^2=f,b^a=bd,c^a=ce,e^a=ef,e^c=ef19 conjugacy classes — view the full character table.
| maximal subgroup of | #1614, #1615, #1617, … |
| maximal normal subgroup of | #1614, #1615, #1617, … |
| maximal quotient of | #1824, #1852, #1872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 165);