Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = #72223000 (order 61440) · Inn(G) = C24 · Out(G) = C2×#11757163⋊C2 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24 |
| commutator, Frattini, cosocle | C2 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,8)(6,14)(9,11)(12,18);(3,4,7,8)(5,11,13,9)(6,10,14,15)(12,17,18,16);(3,5,7,13)(4,9,8,11)(6,12,14,18)(10,16,15,17);(3,6)(4,10)(5,12)(7,14)(8,15)(9,16)(11,17)(13,18)34 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 265);