Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/16) (order 256) · Inn(G) = D8 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D8 |
| commutator, Frattini, cosocle | C8 | C23 |
| socle | C2 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,9)(5,17)(6,21)(7,11)(8,14)(10,19)(13,28)(15,29)(16,27)(18,23)(20,31)(22,26)(24,32)(25,30);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,23)(10,26)(11,25)(12,17)(14,19)(16,30)(20,32)(21,24)(27,28)(29,31);(1,4,7,6)(2,9,11,21)(3,13,15,22)(5,16,20,19)(8,23,25,24)(10,17,27,31)(12,28,29,26)(14,18,30,32)Polycyclic presentation:
2^6|c^2=f,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,d^b=de,e^b=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | #1417, #1418, #2178, … |
| maximal normal subgroup of | #1417, #1418, #2178, … |
| maximal quotient of | #2138, #2158, C4×D16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 189);