Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #33551 (order 256) · Inn(G) = #99 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #99 |
| commutator | C2×C4 | C2×C4 |
| Frattini, cosocle | C2×D4 | C22 |
| prime radical | #127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,2)(3,6,8,5)(4,7);(2,5)(3,8)(6,7)Polycyclic presentation:
2^6|a^2=d,c^2=f,b^a=bc,c^a=ce,e^a=ef,c^b=cf,d^b=def,d^c=df13 conjugacy classes — view the full character table.
| maximal subgroup of | #1402, #1408, #1409, … |
| maximal normal subgroup of | #1402, #1408, #1409, … |
| maximal quotient of | #1318, #1325, #1343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 34);