Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#2010 (order 256) · Inn(G) = C22 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C22 |
| commutator | C2 | C4×C8 |
| socle, prime radical | C22 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6,7,11,8,12,13,18,9,10,14,16,15,17,19,20);(1,2,3,4)(6,10)(11,16)(12,17)(18,20)Polycyclic presentation:
2^6|a^2=c,b^2=d,c^2=e,e^2=f,b^a=bf40 conjugacy classes — view the full character table.
| Frattini subgroup of | #7333, #7389, #7390, … |
| maximal subgroup of | #1313, #1330, #1333, … |
| maximal normal subgroup of | #1313, #1330, #1333, … |
| maximal quotient of | #1311, #1312, #1374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 27);