Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #7622880 (order 512) · Inn(G) = C22⋊C4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊C4 |
| commutator | C22 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| socle | C2 | C2×C22⋊C4 |
| prime radical | C2×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^6|a^2=e,c^2=f,e^2=f,b^a=bd,d^a=df,e^b=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | #1392, #1395, #1757, … |
| maximal normal subgroup of | #1392, #1395, #1757, … |
| maximal quotient of | #1458, #1459, #1464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 94);