Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #23900 (order 256) · Inn(G) = C22⋊C4 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊C4 |
| commutator | C22 | C2×C8 |
| Frattini, cosocle | C22×C4 | C22 |
| prime radical | C23 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,13,12,14,15,16);(1,3,4,8)(2,5,6,7)Polycyclic presentation:
2^6|a^2=d,b^2=e,d^2=f,b^a=bc,c^a=ce,d^b=de22 conjugacy classes — view the full character table.
| maximal subgroup of | #1318, #1319, #1324, … |
| maximal normal subgroup of | #1318, #1319, #1324, … |
| maximal quotient of | #1271, #1280, #1322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 5);