Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C23×#120 (order 256) · Inn(G) = C23 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C23 |
| commutator, socle, prime radical | C22 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,11,13,14,16,15);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,12,14,13,15,16,10)Polycyclic presentation:
2^6|a^2=e,b^2=d,c^2=de,e^2=f,b^a=bd,c^a=cf28 conjugacy classes — view the full character table.
| maximal subgroup of | #1338, #1582, #1584, … |
| maximal normal subgroup of | #1338, #1582, #1584, … |
| maximal quotient of | #1770, #1771, #1916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 127);