Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #511819 (order 512) · Inn(G) = C2×D4 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D4 |
| commutator | C4 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| socle | C2 | C2×C4⋊C4 |
| prime radical | C23 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
51 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7);(1,3,4,12,6,11,5,13)(2,7,8,10,9,15,14,16);(3,11)(7,15)(10,16)(12,13)Polycyclic presentation:
2^6|a^2=e,b^2=d,d^2=f,e^2=f,b^a=bd,d^a=df,c^b=cf22 conjugacy classes — view the full character table.
| maximal subgroup of | #1858, #1861, #1932, … |
| maximal normal subgroup of | #1858, #1861, #1932, … |
| maximal quotient of | #1565, #1569, #1570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 111);