Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = #612424 (order 512) · Inn(G) = C2×D4 · Out(G) = #120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| prime radical | C23 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
74 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)(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14);(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13);(2,4)Polycyclic presentation:
2^6|a^2=f,b^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df19 conjugacy classes — view the full character table.
| maximal subgroup of | #1602, #1603, #1605, … |
| maximal normal subgroup of | #1602, #1603, #1605, … |
| maximal quotient of | #1602, #1605, #1614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 132);