Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×C4×D4 (order 64) · Inn(G) = D4 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D4 |
| commutator | C4 | C2×C8 |
| Frattini, cosocle | C2×C8 | C22 |
| socle | C2 | C22⋊C8 |
| prime radical | #41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,6,11,20,28,7,12,21,29,4,9,18,26)(3,13,15,30,14,24,31,27,16,25,19,32,17,8,22,23);(1,3)(2,8)(4,17)(5,19)(6,14)(7,16)(9,25)(10,27)(11,13)(12,24)(15,21)(18,31)(20,22)(23,29)(26,30)(28,32)Polycyclic presentation:
2^6|a^2=d,c^2=f,d^2=e,e^2=f,b^a=bc,c^b=cf,d^b=df28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#350, #2117, #2170, … |
| maximal normal subgroup of | C2×#350, #2117, #2170, … |
| maximal quotient of | #1315, #1323, D4⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 31);