Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72216662 (order 2048) · Inn(G) = C24 · Out(G) = #3484 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C24 |
| prime radical | C2×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,10)(7,13)(8,16)(18,20);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)Polycyclic presentation:
2^6|b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce22 conjugacy classes — view the full character table.
| Frattini subgroup of | #152139, #152140, #152141, … |
| maximal subgroup of | #1909, #1912, #1915, … |
| maximal normal subgroup of | #1909, #1912, #1915, … |
| maximal quotient of | #2310, #2311, #2317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 217);