Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72215179 (order 1024) · Inn(G) = C24 · Out(G) = C2×#99 |
| 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 | C23 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(3,7)(5,8)(9,10,13,16)(11,14,19,17)(12,20,21,15)(18,23,24,22);(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)Polycyclic presentation:
2^6|a^2=e,b^2=f,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 | #3120, #3123, #3127, … |
| maximal normal subgroup of | #3120, #3123, #3127, … |
| maximal quotient of | #2314, #2320, #2323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 222);