Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #6342371 (order 512) · Inn(G) = C24 · Out(G) = #120 |
| 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 | |
125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,13)(12,15)(14,16);(1,3,5,11)(2,6,7,14)(4,10,12,16)(8,9,15,13);(2,7)(3,10)(6,13)(8,15)(9,14)(11,16);(2,8)(6,9)(7,15)(13,14)Polycyclic presentation:
2^6|b^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce22 conjugacy classes — view the full character table.
| maximal subgroup of | #1600, #1606, #1612, … |
| maximal normal subgroup of | #1600, #1606, #1612, … |
| maximal quotient of | #2312, #2318, #2321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 219);