Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = #394102 (order 512) · Inn(G) = D4 · Out(G) = #392 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4 |
| commutator | C4 | C42 |
| Frattini, cosocle | C22×C4 | C22 |
| socle | C22 | C4⋊C4 |
| prime radical | C23 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(5,6,8,13,10,14,9,12)(7,15,16,11,18,19,17,20);(2,4)(5,7,9,17,10,18,8,16)(6,11,13,19,14,20,12,15)Polycyclic presentation:
2^6|a^2=d,b^2=e,c^2=f,d^2=f,e^2=f,b^a=bc,c^a=cf,e^a=ef28 conjugacy classes — view the full character table.
| maximal subgroup of | #1349, #1350, C2×#341, … |
| maximal normal subgroup of | #1349, #1350, C2×#341, … |
| maximal quotient of | #1277, #1278, #1287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 22);