Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| 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 | #3511 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
447 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,6,9)(3,10,13,15)(4,8,5,14)(7,12,16,11)(17,18,22,26)(19,27,30,31)(20,23,21,25)(24,29,32,28);(1,3,6,13)(2,7,9,16)(4,11,5,12)(8,15,14,10)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,16,10)(17,19,22,30)(18,24,26,32)(20,28,21,29)(23,31,25,27);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,10,16,15)(17,20,22,21)(18,25,26,23)(19,28,30,29)(24,27,32,31)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #249806, #249808, #249810, … |
| maximal normal subgroup of | #249806, #249808, #249810, … |
| maximal quotient of | #168932, #168987, #169032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11821);