Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 | |
353 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 147 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:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,21,23,33)(12,26,27,34)(14,20,15,19)(17,25,30,24)(18,29,31,28)(22,36,37,32)(35,39,40,38);(1,3,4,8)(2,5,6,7)(10,16)(17,30)(18,31)(19,20)(21,33)(26,34)(32,36)(35,40);(9,11,14,24,13,23,15,25)(10,17,19,21,16,30,20,33)(12,22,28,38,27,37,29,39)(18,32,26,35,31,36,34,40);(9,12)(10,18)(11,22)(13,27)(14,28)(15,29)(16,31)(17,32)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(30,36)(33,40)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138098, #138099, #138110, … |
| maximal normal subgroup of | #138098, #138099, #138110, … |
| maximal quotient of | #352402, #352403, #352408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26209);