Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = C24×#120 (order 512) · Inn(G) = C24 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C24 |
| prime radical | #121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(11,15)(13,16);(1,2,4,6)(3,5,8,7)(9,10,12,14)(11,13,15,16);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,15)(14,16);(1,3,4,8)(2,5,6,7)Polycyclic presentation:
2^6|b^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce22 conjugacy classes — view the full character table.
| maximal subgroup of | #3118, #3121, #3124, … |
| maximal normal subgroup of | #3118, #3121, #3124, … |
| maximal quotient of | #2314, #2319, #2320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 223);