Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,24)(13,17)(14,19)(15,22)(16,28)(18,26)(21,37)(23,31)(25,30)(27,35)(29,34)(32,38)(33,39)(36,40);(1,3,5,4)(2,6,8,7)(9,11,14,23,13,22,26,38)(10,15,19,32,17,20,18,31)(12,25,29,39,27,40,28,21)(16,33,24,36,34,37,35,30);(1,3,5,4)(2,7,8,6)(9,12,26,29,13,27,14,28)(10,16,19,35,17,34,18,24)(11,21,38,25,22,39,23,40)(15,30,32,37,20,36,31,33);(1,4,5,3)(2,7,8,6)(10,17)(15,20)(16,34)(18,19)(24,35)(30,36)(31,32)(33,37)Polycyclic presentation:
2^8|b^2=egh,c^2=ef,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #143841, #143842, #143845, … |
| maximal normal subgroup of | #143841, #143842, #143845, … |
| maximal quotient of | #325330, #325336, #325341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21722);