Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72215688 (order 2048) · Inn(G) = C24 · Out(G) = #2196 |
| 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 | |
103 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)(9,10,12,14)(11,15,16,13);(1,2,4,6)(3,5,8,7);(1,2)(3,5)(4,6)(7,8)(9,11,12,16)(10,13,14,15);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^6|a^2=f,b^2=e,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce22 conjugacy classes — view the full character table.
| maximal subgroup of | #1599, #1605, #1611, … |
| maximal normal subgroup of | #1599, #1605, #1611, … |
| maximal quotient of | #2316, #2317, #2329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 225);