Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = #6628842 (order 512) · Inn(G) = D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D4 |
| commutator | C4 | C42 |
| Frattini, cosocle | C22×C4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,4)(5,6,7,8)(10,14)(11,12)(15,16);(2,4)(9,10,12,15,13,16,11,14)Polycyclic presentation:
2^6|a^2=d,b^2=e,c^2=f,e^2=f,b^a=bc,c^a=cf,e^a=ef28 conjugacy classes — view the full character table.
| maximal subgroup of | #1347, #1348, C2×#340, … |
| maximal normal subgroup of | #1347, #1348, C2×#340, … |
| maximal quotient of | #1277, #1279, #1286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 21);