Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | Aut(G) order 16384 |
| 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 | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 123 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)(3,6)(4,5)(7,8)(9,10,16,22)(11,18,25,32)(12,17,24,33)(13,27,30,35)(14,21,15,20)(19,29,36,28)(23,37,39,31)(26,34,38,40);(2,5)(6,8)(9,11,14,12,16,25,15,24)(10,17,20,18,22,33,21,32)(13,23,28,26,30,39,29,38)(19,31,27,34,36,37,35,40);(1,3,4,7)(2,6,5,8)(9,12,14,25,16,24,15,11)(10,18,20,33,22,32,21,17)(13,26,28,39,30,38,29,23)(19,34,27,37,36,40,35,31);(9,13)(10,19)(11,23)(12,26)(14,28)(15,29)(16,30)(17,31)(18,34)(20,27)(21,35)(22,36)(24,38)(25,39)(32,40)(33,37)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=ch,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #138087, #138093, #273729, … |
| maximal normal subgroup of | #138087, #138093, #273729, … |
| maximal quotient of | #350994, #350995, #350998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26053);