Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19420 |
| 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 | C2×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
293 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 293 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:
(1,2,5,8)(3,9,11,16)(4,13,10,7)(6,14,15,12)(17,18,22,26)(19,23,28,32)(20,25,21,30)(24,29,31,27);(1,3,10,14,5,11,4,12)(2,6,13,16,8,15,7,9)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(3,11)(6,15)(9,16)(12,14)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #272039, #272040, #272041, … |
| maximal normal subgroup of | #272039, #272040, #272041, … |
| maximal quotient of | #189148, #197874, #197956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12441);