Finite Group DatabasebetaFinite Group Databasebeta| order | 64 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = D4×AGL(1,ℤ/8) (order 256) · Inn(G) = #121 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23 |
| prime radical | #41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12)(3,13,17,23)(4,18,20,26)(5,11,6,10)(8,16,24,15)(9,21,25,19)(14,28,30,22)(27,31,32,29);(1,3,6,16,7,17,5,15)(2,8,11,23,12,24,10,13)(4,14,21,31,20,30,19,29)(9,22,26,32,25,28,18,27);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,26)(12,25)(13,27)(15,29)(16,31)(17,30)(23,32)(24,28)Polycyclic presentation:
2^6|a^2=f,b^2=df,d^2=f,e^2=f,b^a=bd,c^a=ce,d^a=df,e^a=ef,e^c=ef16 conjugacy classes — view the full character table.
| maximal subgroup of | #2194, #2195, #2222, … |
| maximal normal subgroup of | #2194, #2195, #2222, … |
| maximal quotient of | #1669, #1670, #1673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(64, 154);