Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10518 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
191 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,8,4)(5,12,15,9)(7,14,16,13)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31);(1,3,6,8)(2,7,10,16)(4,12,11,9)(5,14,15,13)(17,19)(18,23)(20,29)(21,30)(22,28)(24,32)(25,27)(26,31);(1,4,5,7,6,11,15,16)(2,8,9,14,10,3,12,13)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=dh,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,e^c=egh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93820, #93855, #123147, … |
| maximal normal subgroup of | #93820, #93855, #123147, … |
| maximal quotient of | #104862, #104867, #104872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3511);