Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C24×#120 (order 512) · Inn(G) = C24 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,4)(5,6,9,10)(7,11,12,8);(6,10)(8,11);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11);(5,8)(6,7)(9,11)(10,12)Polycyclic presentation:
2^6|a^2=e,b^a=be,c^a=cf,d^b=de25 conjugacy classes — view the full character table.
| maximal subgroup of | #3264, #3266, #3267, … |
| maximal normal subgroup of | #3264, #3266, #3267, … |
| maximal quotient of | D4×C22⋊C4, #2339, #2342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 227);