Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,6)(3,7,8,5)(11,16)(13,15)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(18,22)(23,31)(24,29)(25,32);(10,13)(11,16)(14,15)(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=dg,c^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fh,e^c=eg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94713, #151380, #151397, … |
| maximal normal subgroup of | #94713, #151380, #151397, … |
| maximal quotient of | #104720, #104724, #104726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3340);