Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C25×D4 (order 256) · Inn(G) = C2×D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| prime radical | C2×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,13)(8,10)(9,17)(11,15)(12,16)(14,20)(18,19);(1,2,3,4)(6,10)(11,19)(12,17)(13,20);(5,7,8,14)(6,11,10,19)(9,18,16,15)(12,20,17,13)Polycyclic presentation:
2^6|b^2=df,c^2=f,e^2=f,b^a=bd,c^a=ce,e^a=ef,e^c=ef19 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#464, #3035, #3037, … |
| maximal normal subgroup of | C2×#464, #3035, #3037, … |
| maximal quotient of | #1678, #1680, #1682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 145);