Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#23900 (order 512) · Inn(G) = C22 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22 |
| commutator | C2 | C2×C42 |
| Frattini, cosocle | C2×C4 | C23 |
| socle | C22 | C22×C4 |
| prime radical | C23 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
65 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20);(7,13)(10,16)(14,19)(17,20)Polycyclic presentation:
2^6|a^2=d,b^2=e,d^2=f,c^b=cf40 conjugacy classes — view the full character table.
| derived subgroup of | #143170, #143171, #143172, … |
| Frattini subgroup of | #7239, #7240, C2×#7239, … |
| maximal subgroup of | #1378, #1379, #1747, … |
| maximal normal subgroup of | #1378, #1379, #1747, … |
| maximal quotient of | C8×M16, #1450, #1451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 86);