Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#6342371 (order 1024) · Inn(G) = C23 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C23 |
| commutator | C22 | C22×C4 |
| prime radical | C22×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6,7,8)(10,12);(5,6)(7,8)(9,10,11,12);(1,2)(3,4)(5,6)(7,8)Polycyclic presentation:
2^6|a^2=f,b^2=df,b^a=bd,c^a=ce28 conjugacy classes — view the full character table.
| maximal subgroup of | #1390, #1790, #1867, … |
| maximal normal subgroup of | #1390, #1790, #1867, … |
| maximal quotient of | C22⋊#95, #1436, #1437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 67);