Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72216213 (order 6144) · Inn(G) = C24 · Out(G) = #4625⋊C2 |
| 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 | C24 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,8)(3,9,11,7)(4,6,12,14)(10,15,16,13);(2,6)(3,10)(8,14)(11,16);(1,3,5,11)(2,7,8,9)(4,10,12,16)(6,13,14,15);(2,6)(7,13)(8,14)(9,15)Polycyclic presentation:
2^6|a^2=f,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce22 conjugacy classes — view the full character table.
| derived subgroup of | #543⋊C3, #2029⋊C3, #2030⋊C3, … |
| Fitting subgroup of | #543⋊C3, #85548, #85549, … |
| maximal subgroup of | #1596, #1602, #1608, … |
| maximal normal subgroup of | #1596, #1602, #1608, … |
| maximal quotient of | #2315, #2329, #2437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 224);