Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C2×#2196 (order 256) · Inn(G) = #120 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)Polycyclic presentation:
2^6|c^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=ef,d^c=df16 conjugacy classes — view the full character table.
| maximal subgroup of | #1404, #2202, #2203, … |
| maximal normal subgroup of | #1404, #2202, #2203, … |
| maximal quotient of | #1597, #1598, #1599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 135);