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 |
| prime radical | H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14);(4,14)(7,12)(9,16)(10,13)Polycyclic presentation:
2^6|a^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 | #1403, #2197, #2199, … |
| maximal normal subgroup of | #1403, #2197, #2199, … |
| maximal quotient of | #1601, #1602, #1606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 136);