Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#72210665 (order 12288) · Inn(G) = C22 · Out(G) = C2×#72209884 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22 |
| commutator | C2 | C2×C42 |
| Frattini, socle, prime radical, cosocle | C23 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
89 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4)(9,10)(11,14)(12,13)(15,16);(5,6,7,8)(10,13)(14,16);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^6|a^2=e,b^2=f,c^2=d,b^a=bd40 conjugacy classes — view the full character table.
| derived subgroup of | #94634, #94635, #94646, … |
| Frattini subgroup of | #7115, #7129, C2×#7115, … |
| Fitting subgroup of | #376⋊C3, #67997, #72986, … |
| maximal subgroup of | #1274, #1300, #1500, … |
| maximal normal subgroup of | #1274, #1300, #1500, … |
| maximal quotient of | C4×#95, #1434, #1744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 57);