Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C22×AGL(1,ℤ/8) (order 128) · Inn(G) = D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D4 |
| commutator | C4 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| socle | C2 | C4×D4 |
| prime radical | #135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,9,6,3,10,8)(4,7,12,15,13,11,16,14);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,14)(15,16);(1,4,5,12,6,13,10,16)(2,7,9,15,3,11,8,14)Polycyclic presentation:
2^6|a^2=e,c^2=e,d^2=f,e^2=f,b^a=bd,d^a=df,d^b=df28 conjugacy classes — view the full character table.
| maximal subgroup of | #1335, #1336, #2170, … |
| maximal normal subgroup of | #1335, #1336, #2170, … |
| maximal quotient of | C8×D8, C8×SD16, C8×Q16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 124);