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) = C24 |
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 | |
119 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)(3,4)(5,6)(7,8)(10,13)(14,15);(2,6)(4,7)(9,10,12,13)(11,15,16,14);(1,3)(2,7)(4,6)(5,8);(1,4,5,7)(2,3,6,8)(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^6|b^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de25 conjugacy classes — view the full character table.
| maximal subgroup of | #3297, #3301, #3307, … |
| maximal normal subgroup of | #3297, #3301, #3307, … |
| maximal quotient of | #2345, #2347, #2352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 234);