Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = C22×D42 (order 256) · Inn(G) = C22⋊C4 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C22⋊C4 |
| commutator | C2×C4 | C2×C4 |
| Frattini, cosocle | C42 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,5)(3,7)(6,8)(9,10,12,14,13,16,11,15);(1,2,4,6)(3,8,7,5)(10,14)(11,12)(15,16)Polycyclic presentation:
2^6|a^2=d,b^2=e,c^2=f,d^2=ef,b^a=bc,c^a=ce,c^b=cf,d^b=def19 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#332, #1540, #1542, … |
| maximal normal subgroup of | C2×#332, #1540, #1542, … |
| maximal quotient of | #1271, #1296, #1363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 13);