Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = #72218858 (order 23040) · Inn(G) = C24 · Out(G) = C2×S6 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24 |
| commutator, Frattini, socle, cosocle | C2 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(3,10)(6,9)(11,16)(13,15);(2,7)(6,9)(8,14)(13,15);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)Polycyclic presentation:
2^6|e^2=f,b^a=bf,d^a=df,c^b=cf34 conjugacy classes — view the full character table.
| derived subgroup of | #32894⋊C3, #32895⋊C3, #32896⋊C3, … |
| Fitting subgroup of | #5945, #5950, #585⋊C5, … |
| maximal subgroup of | #1791, #1794, #2901, … |
| maximal normal subgroup of | #1791, #1794, #2901, … |
| maximal quotient of | #3428, C4×H(5,𝔽2), C4×2−1+4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 266);