Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| 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 | #3106 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
306 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 109 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)(3,10,12,15)(4,8,5,14)(7,13,16,11)(17,18,22,27)(19,28,20,24)(21,26,23,31)(25,29,32,30);(1,3,4,13,6,12,5,11)(2,7,14,15,9,16,8,10)(17,19,22,20)(18,24,27,28)(21,30,23,29)(25,26,32,31);(17,20,22,19)(18,25,27,32)(21,30,23,29)(24,26,28,31);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(17,21,22,23)(18,26,27,31)(19,29,20,30)(24,25,28,32)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268630, #268633, #268636, … |
| maximal normal subgroup of | #268630, #268633, #268636, … |
| maximal quotient of | #189161, #197895, #197976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12491);