Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#575 (order 128) · Inn(G) = D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D4 |
| commutator | C4 | C2×C8 |
| Frattini, cosocle | C2×C8 | C22 |
| socle | C2 | C4⋊C8 |
| prime radical | C22 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
23 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,5,10,6,9,14,16,7,11,13,8,4,15,3,12);(1,3,4,13,7,14,6,5)(2,8,9,12,11,10,15,16)Polycyclic presentation:
2^6|a^2=d,b^2=c,c^2=f,d^2=e,e^2=f,b^a=bc,c^a=cf28 conjugacy classes — view the full character table.
| maximal subgroup of | #1335, #1336, #1357, … |
| maximal normal subgroup of | #1335, #1336, #1357, … |
| maximal quotient of | #1367, #1370, #1376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 45);