Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #7622880 (order 512) · Inn(G) = C22⋊C4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊C4 |
| commutator | C22 | C22×C4 |
| Frattini, cosocle | C23 | C23 |
| socle | C2 | C2×C22⋊C4 |
| prime radical | C2×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)Polycyclic presentation:
2^6|a^2=e,c^2=f,b^a=bd,d^a=df,e^b=ef22 conjugacy classes — view the full character table.
| maximal subgroup of | #1391, #1395, #1757, … |
| maximal normal subgroup of | #1391, #1395, #1757, … |
| maximal quotient of | #1458, #1460, #1461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 91);