Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = C25×D4 (order 256) · Inn(G) = C2×D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| prime radical | #121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(3,4)(6,9)(7,11)(10,12);(5,6,7,9,8,10,11,12);(2,4)(6,10)(9,12)Polycyclic presentation:
2^6|b^2=d,d^2=f,b^a=bd,c^a=ce,d^a=df,c^b=cf16 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#468, #3054, #3055, … |
| maximal normal subgroup of | C2×#468, #3054, #3055, … |
| maximal quotient of | #1685, #1686, #1691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 149);