Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = #7630939 (order 512) · Inn(G) = D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4 |
| commutator | C4 | C22×C4 |
| Frattini, cosocle | C2×C4 | C23 |
| socle | C22 | C2×D4 |
| prime radical | C2×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)Polycyclic presentation:
2^6|a^2=e,c^2=f,d^2=f,b^a=bd,d^a=df,d^b=df28 conjugacy classes — view the full character table.
| maximal subgroup of | #1764, #1765, #1794, … |
| maximal normal subgroup of | #1764, #1765, #1794, … |
| maximal quotient of | #1476, #1482, #1488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 97);