Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #511819 (order 512) · Inn(G) = C42 · Out(G) = #120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
| unitriangular | UT(3,ℤ/4) |
| Heisenberg | H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, commutator | C4 | C42 |
| Frattini, cosocle | C22×C4 | C22 |
| socle | C2 | #95 |
| prime radical | C23 | D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)Polycyclic presentation:
2^6|a^2=d,b^2=e,c^2=f,b^a=bc,e^a=ef,d^b=df22 conjugacy classes — view the full character table.
| maximal subgroup of | #1339, #1340, C2×H(3,ℤ/4), … |
| maximal normal subgroup of | #1339, #1340, C2×H(3,ℤ/4), … |
| maximal quotient of | #1274, #1275, #1284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 18);