Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) = #72209712 (order 1536) · Inn(G) = C24 · Out(G) = #1020 |
| 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 | |
126 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)(3,10)(4,9)(5,7)(6,12)(8,14)(11,15)(13,16);(1,3)(2,6)(4,12)(5,13)(7,15)(8,16)(9,10)(11,14);(2,7)(3,11)(6,10)(8,9)(12,13)(15,16);(2,8)(3,12)(6,15)(7,9)(10,16)(11,13)Polycyclic presentation:
2^6|b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de19 conjugacy classes — view the full character table.
| Fitting subgroup of | #560⋊C3, #4626⋊C2 |
| maximal subgroup of | #2200, C2×#560, #3529, … |
| maximal normal subgroup of | #2200, C2×#560, #3529, … |
| maximal quotient of | #2375, #2376, #2377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 241);