Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) = #72210547 (order 1024) · Inn(G) = C24 · Out(G) = AΣL(1,𝔽16) |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5)(3,7)(9,10)(11,15)(12,13)(14,16);(2,5)(6,8)(10,13)(14,15);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,14,13,15);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^6|c^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de19 conjugacy classes — view the full character table.
| maximal subgroup of | #1656, #1661, #1662, … |
| maximal normal subgroup of | #1656, #1661, #1662, … |
| maximal quotient of | #2376, #2377, #2378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 243);