Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #3025 (order 128) · Inn(G) = C22⋊C4 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊C4 |
| commutator | C22 | C2×C8 |
| Frattini, cosocle | C2×C8 | C22 |
| socle | C2 | C22⋊C8 |
| prime radical | C23 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
29 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16);(2,6)(3,9)(8,13)(10,15)Polycyclic presentation:
2^6|a^2=d,d^2=e,e^2=f,b^a=bc,c^a=cf,d^b=df22 conjugacy classes — view the full character table.
| maximal subgroup of | #1320, #1321, #1327, … |
| maximal normal subgroup of | #1320, #1321, #1327, … |
| maximal quotient of | #1314, #1322, #1330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 30);