Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = #72217027 (order 1536) · Inn(G) = C24 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4 | C24 |
| commutator, socle | C2 | C23×C4 |
| prime radical | C2×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16);(2,7)(6,12)(8,14)(13,16)Polycyclic presentation:
2^6|a^2=e,e^2=f,d^a=df,c^b=cf34 conjugacy classes — view the full character table.
| Fitting subgroup of | #568⋊C3, #73129, #73130, … |
| maximal subgroup of | #1858, #1861, #1868, … |
| maximal normal subgroup of | #1858, #1861, #1868, … |
| maximal quotient of | #2873, #2875, #2879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 249);