Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19504 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
288 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 105 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)(3,6,10,14)(4,7,12,8)(11,13,16,15)(18,22)(19,26)(20,23)(25,30)(27,28)(29,31);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,18,23,30,21,25,20,22)(19,24,26,31,27,32,28,29);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,28)(21,27)(22,29)(23,26)(25,32)(30,31);(3,10)(6,14)(11,16)(13,15)(18,25)(19,27)(22,30)(26,28)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,d^c=dg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19515, #6354603, #6354643, … |
| maximal normal subgroup of | C2×#19515, #6354603, #6354643, … |
| maximal quotient of | #189615, #197916, #197997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12503);