Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = C22×#120 (order 128) · Inn(G) = C22⋊C4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊C4 |
| commutator | C2×C4 | C2×C4 |
| Frattini, cosocle | C22×C4 | C22 |
| prime radical | #121 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,6,7,14)(3,10)(8,15,9,11)(12,16);(1,2)(3,11)(4,8)(5,9)(6,12)(7,13)(10,14)(15,16)Polycyclic presentation:
2^6|a^2=d,c^2=f,b^a=bc,c^a=ce,c^b=cf,d^b=def19 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#327, #1500, #1501, … |
| maximal normal subgroup of | C2×#327, #1500, #1501, … |
| maximal quotient of | #1270, #1271, #1294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 8);