Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#72210665 (order 12288) · Inn(G) = C23 · Out(G) = C2×#11714279 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C23 |
| commutator | C22 | C22×C4 |
| prime radical | C25 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6,7,8)(10,12);(9,10)(11,12);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)Polycyclic presentation:
2^6|a^2=f,b^a=bd,c^a=ce28 conjugacy classes — view the full character table.
| derived subgroup of | #94626, #94627, #94628, … |
| Frattini subgroup of | #7338, #7349, #7394, … |
| Fitting subgroup of | #379⋊C3, #67998, #72989, … |
| maximal subgroup of | #1304, #1316, #1781, … |
| maximal normal subgroup of | #1304, #1316, #1781, … |
| maximal quotient of | C22⋊#95, #1438, #1446, … |
| outer automorphism group of | #870, #2090, #4690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 60);