Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = #392109 (order 512) · Inn(G) = C2×D4 · Out(G) = #120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| prime radical | C2×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24);(2,5)(6,8)(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19);(1,3,4,7)(2,6,5,8)(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24)Polycyclic presentation:
2^6|b^2=ef,c^2=d,d^2=f,e^2=f,b^a=bd,c^a=ce,d^a=df,e^a=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | #1414, C2×#488, #3131, … |
| maximal normal subgroup of | #1414, C2×#488, #3131, … |
| maximal quotient of | #1835, #1933, #1934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 169);