Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3552 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
439 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 439 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9)(3,7,10,14)(4,6,12,8)(11,13,16,15)(18,22)(19,25)(20,27)(24,28)(26,29)(30,32);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,18,21,24)(19,23,26,31)(20,28,27,22)(25,30,29,32);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,19)(18,23)(20,29)(21,26)(22,30)(24,31)(25,27)(28,32);(3,10)(7,14)(11,16)(13,15)(19,26)(23,31)(25,29)(30,32)Polycyclic presentation:
2^8|a^2=gh,b^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #347186, #347187, #347188, … |
| maximal normal subgroup of | #347186, #347187, #347188, … |
| maximal quotient of | #167779, #167804, #167830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11688);