Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C2×#6342371 (order 1024) · Inn(G) = C23 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(7,11)(9,12)(13,14,16,18)(15,19,20,17);(2,4)(5,6,8,10)(7,12,11,9)(13,14)(15,19)(16,18)(17,20);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^6|a^2=d,b^2=de,c^2=d,b^a=bd,c^a=ce,c^b=cf22 conjugacy classes — view the full character table.
| maximal subgroup of | #1353, #1615, #1616, … |
| maximal normal subgroup of | #1353, #1615, #1616, … |
| maximal quotient of | #1441, #1442, #1443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 81);