Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10362 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10,12,16)(4,8)(5,14)(6,9)(7,13,15,11)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27);(3,11,12,13)(7,10,15,16)(18,22)(23,31)(24,29)(25,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,16)(9,15)(10,14)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25)Polycyclic presentation:
2^8|a^2=d,b^2=dh,c^2=h,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eh,f^a=fg,c^b=cf,d^c=dg,e^c=egh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10367, #237946, #237947, … |
| maximal normal subgroup of | C2×#10367, #237946, #237947, … |
| maximal quotient of | #104742, #104744, #104746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3355);