Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #234591 (order 512) · Inn(G) = C22 · Out(G) = #2021 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C22 |
| commutator | C2 | C4×C8 |
| socle, prime radical | C22 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,6,4,5,7,8);(2,5)(6,8)(9,10,11,13,12,14,15,16)Polycyclic presentation:
2^6|a^2=c,b^2=d,c^2=e,d^2=f,b^a=be40 conjugacy classes — view the full character table.
| maximal subgroup of | #1313, #1336, C2×#322, … |
| maximal normal subgroup of | #1313, #1336, C2×#322, … |
| maximal quotient of | #1273, #1312, #1313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 3);