Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C24×#120 (order 512) · Inn(G) = C23 · Out(G) = C26 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23 |
| commutator, Frattini, socle, cosocle | C22 | C24 |
| prime radical | #121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16);(2,5)(6,8)(9,11,12,15)(10,13,14,16);(10,14)(13,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^6|b^2=f,d^2=e,b^a=be,c^a=cf28 conjugacy classes — view the full character table.
| maximal subgroup of | #1482, #1483, #1494, … |
| maximal normal subgroup of | #1482, #1483, #1494, … |
| maximal quotient of | #2283, #2290, #2293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 210);